=Paper=
{{Paper
|id=Vol-1424/Paper4
|storemode=property
|title=Type-2 Fuzzy Uncertainty in Goal Programming
|pdfUrl=https://ceur-ws.org/Vol-1424/Paper4.pdf
|volume=Vol-1424
|dblpUrl=https://dblp.org/rec/conf/ijcai/Patino-Callejas15
}}
==Type-2 Fuzzy Uncertainty in Goal Programming==
https://ceur-ws.org/Vol-1424/Paper4.pdf
Type-2 Fuzzy Uncertainty in Goal Programming
Juan S. Patiño-Callejas, Krisna Y. Espinosa-Ayala, and Juan C. Figueroa-Garcı́a
Universidad Distrital Francisco José de Caldas
Bogotá - Colombia
juansebastianpatinoc@gmail.com, joespinosa1018@gmail.com, jcfigueroa@udistrital.edu.co
Abstract Section 6 shows an application example; and finally Section
7 presents the concluding remarks of the study.
This paper presents a goal programming model for
problems where resources are defined by the opin-
ion of multiple experts. Through the use of Type-2 2 Basic on Fuzzy sets
fuzzy sets, we propose a model that includes human According to Klir and Yuan [Klir and Yuan, 1995], a fuzzy set
being like information in order to define the pa- is a function A : X → [0, 1]. The notation µA is equivalent
rameters of a goal programming problem, and then to describe the membership function µ that describes A, this
solve it using a constructive approach that uses LP is µA : X → [0, 1] where x ∈ X is the universe of discourse
models due to its efficiency. An application exam- over A is defined, as follows:
ple is provided and explained, and some concluding A : X → [0, 1]
remarks are provided.
A = (x, µA (x)) : x ∈ X (1)
1 Introduction 2.1 Type-2 Fuzzy Sets
Decision making in practical applications has to face human A Type-2 Fuzzy set, Mendel [Mendel, 2001] is an ordered
being interaction and social aspects. Some situations have pair {(x, µÃ (x)) : x ∈ X}, where A is a linguistic label Ã
to solve multiple goals involving multiple people that try to represents the uncertainty about the word A. And its mathe-
solve the same problem with different objectives. To solve matical definition is:
those problems, goal programming offers an efficient tool to
find a solution. Ã : X → F [0, 1]
A common situation in applied goal programming includes à = (x, µÃ (x)) : x ∈ X (2)
multiple experts and uncertainty around the exact value of Z Z
a desired goal, where fuzzy sets appear as a useful tool for à = fx (u) /(x, u), Jx ⊆ [0, 1]
handling uncertainty coming from different people. Classical x∈X u∈Jx
fuzzy goal programming has been proposed by Narasimhan where fx (u) /u is a secondary membership function of à on
[Narasimhan, 1980], and later developed by Yang [Yang et x ∈ X and u is the domain of uncertainty.
al., 1991], Turgay & Taşkın [Safiye and Harun, 2014], Li
& Gang [Li, 2012],Hu, Zhang & Wang [Hu et al., 2014],
Khalili-Damghani & Sadi-Nezhad [Khalili-Damghani et al., Why Fuzzy Sets? Fuzzy sets has the property of handling
2013], in both theoretical and practical situations. uncertainty coming from human knowledge, which com-
Using the results of Narasimhan [Narasimhan, 1980], Yang monly appear in decision making. In the case of numerical
[Yang et al., 1991] has designed a smaller model (in terms of uncertainty, fuzzy sets handle imprecision about X that ap-
amount of variables) that leads to the same solution. In this pears in cease where no historical/statistical data is available,
paper we propose to extend the classical goal programming so the only way to estimate parameters and/or variables is by
problem to a case where multiple experts deal with multiple using approximate information coming from the experts of
goals by using Type-2 fuzzy sets and α-cuts to handle lin- the problem that can be represented through fuzzy numbers.
guistic/numerical uncertainty coming from experts and Lin- 2.2 α-cuts
ear Programming (LP) methods for handling goal program-
ming. One of the most used ways to decompose A is through α-cuts.
The paper is organized as follows: Section 1 introduces The α-cut of a A, namely αA, is defined as:
the main problem. Section 2 presents some basics on fuzzy α
A = {x ∈ X : µA (x) > α}, (3)
sets. In Section 3, goal programming LP model is referred. S
Section 4 presents the Yang [Yang et al., 1991] proposal for Thus, a fuzzy set A is the union of its α-cuts, α∈[0,1] α ·
α
fuzzy goal programming. Section 5 contains the proposal; A, where ∪ denotes union [Klir and Yuan, 1995]. Now, the
extension of α-cut of A to the α-cut of à (see [Figueroa- is
Garcı́a et al., 2015]) allows us to say that the primary α-cut
n
of an Interval Type-2 fuzzy set α Ã is the union of all x ∈ X X
whose primary memberships Jx are greater than α, Jx > α, min dk1 + dk2
k
this is: k=1
s.t.
α
à = {x ∈ X : µÃ (x, u) > α; u ∈ Jx ⊆ [0, 1]}, (4)
Ak x + dk1 − dk2 ∼
= B̃k , (7)
3 Goal programming A′k x ≤ Bk′
x, dk1 , dk2 ≥ 0; ∀ k,
Charnes, Cooper & Wagner [Charnes and Cooper, 1961;
1977] has proposed an LP model that tries to minimize devi-
ations from different goals (desired objectives) through min- where B̃k ∈ F1 the fuzzy aspiration level, dk1 , dk2 ∈ R are
imizing the absolute deviations dk of the constraints of the negative and positive deviations from the goal bk , Ak is the
problem Ak x regarding its desired value Bk e g. mink {D = set of n constraints related to fuzzy goals, A′k is a set of crisp
P n constraints of the problem, Bk′ is its set of boundaries, and
k=1 |Ak x − Bk |}. This model is equivalent to the follow-
ing LP model (see Charnes, Cooper & Wagner [Charnes and x ∈ Rm is the set of decision variables of the problem.
Cooper, 1961; 1977]): Every Type-2 fuzzy goal is defined by its LMF and UMF,
n
as shown as follows:
X
min dk1 + dk2
k 0 if Gk (x) ≤ bk + bk2 ,
k=1
G (x) − bk
k
s.t. 1− , if bk ≤ Gk (x) ≤ bk + bk2 ,
bk2
Ak x + dk1 − dk2 = Bk ,
(5) µb̃k = 1 if Gk (x) = bk ,
A′k x ≤ Bk′
b k − Gk (x)
1− , if bk − bk1 ≤ Gk (x) ≤ bk ,
x, dk1 , dk2 ≥ 0; ∀ k,
b k1
0 otherwise,
where Bk ∈ R is the aspiration level, dk1 , dk2 ∈ R are nega- (8)
tive and positive deviations from the goal Bk , Ak is the set of
n constraints related to goals, A′k is a set of crisp constraints
0 if Gk (x) ≤ bk + bk2 ,
of the problem, Bk′ is its set of boundaries, and x ∈ Rm is the
G (x) − bk
k
set of decision variables of the problem. A negative deviation 1− , if bk ≤ Gk (x) ≤ bk + bk2 ,
bk2
quantifies a lack of satisfaction of the desired aspiration level,
and a positive deviation quantifies an excess over the desired µb̃ = 1 if Gk (x) = bk ,
k
bk − Gk (x)
aspiration level.
1− , if bk − bk1 ≤ Gk (x) ≤ bk ,
bk1
0 otherwise,
4 Fuzzy Goal Programming
(9)
Fuzzy goal programming has been proposed by Narasimhan where µ defines the LMF of the kth goal, and µ defines the
[Narasimhan, 1980], Narasimhan & Hanna [Hannan, 1981], UMF of the kth goal. A graphical display of a Type-2 fuzzy
and Yang [Yang et al., 1991] has proposed a smaller model goal is shown in Figure 1.
that obtains an equivalent solution that the presented by
[Narasimhan, 1980; Hannan, 1981]. Yang’s proposal defines
the membership function of the kth fuzzy goal Bk namely
µBk , as follows:
0 if Gk (x) ≤ bk + bk2 ,
G (x) − b
k k
1− , if bk ≤ Gk (x) ≤ bk + bk2 ,
bk2
µBk = 1 if Gk (x) = bk ,
b − G (x)
k k
1− , if bk − bk1 ≤ Gk (x) ≤ bk ,
bk1
0 otherwise,
(6)
where k ∈ n denotes the kth goal, Gk (x) is the kth constraint
to be fulfilled, bk ∈ R is the aspiration level of the kth goal, Figure 1: Interval Type-2 fuzzy goal B̃k
and dk1 and dk2 are the maximum negative and positive de-
viations from bk , respectively. Then the resulting LP model
4.1 The Proposal Then from the k goal values the value of the deviations in
Our proposal extends the classical fuzzy goal programming the linear goal programming problem (7) are computed, as
model to a Type-2 fuzzy environment, as follows: a four-step LP method which finds the following crisp solu-
Xn tions:
min dk1 + dk2 α
B̌k,l → αžl (15)
k
k=1 α α
B̂k,l → ẑl (16)
s.t.
α α
B̌k,r → žr (17)
Ak x + dk1 − dk2 ≈ B̃k , (10)
α α
A′k x ≤ Bk′ B̂k,r → ẑr (18)
x, dk1 , dk2 ≥ 0; ∀ k, Now, every set of goals αB̌k,l , αB̂k,l , αB̌k,r , αB̂k,r has to be
where B̃k ∈ R is a Type-2 fuzzy aspiration level, dk1 , dk2 ∈ solved using (7). This way, the set of Type-2 fuzzy goals B̃
R are negative and positive deviations from the goal B̃k , Ak leads to a set of optimal solutions ž, as follows:
is the set of n constraints related to goals, A′k is a set of crisp f
constraints of the problem, Bk′ is its set of boundaries, and B̃ −−→ z̃ (19)
x ∈ Rm is the set of decision variables of the problem. where f is a function, in this case an LP method.
The proposed approach to find a solution of the problem
is by using a constructive method based on α-cuts which ba- 6 Experimentation and results
sically decomposes B̃k into α-cuts and find a crisp solution As application example we use the proposed by [Narasimhan,
for every of the 4 boundaries of every α-cut. The method is 1980] and extended by [Chen and Tsai, 2001] which is com-
described as follows. posed by three fuzzy goals, as shown as follows:
5 α-cuts and deviations in Fuzzy Goal G1 : 80x1 + 40x2 = ∼ 630,
Programming ∼
G2 : x1 = 7, (20)
There is a relationship between satisfaction levels, α-cuts, G3 : x2 = ∼ 4,
and the goal value. It is clear that there exists a set X that where x1 and x2 are the manufacturing quantities of two
satisfies every α-cut which leads to two intervals, one for the products which regard to three goals: G1 is a profit goal, and
left side [αB̂k,l ,α B̌k,l ] and one for the right side [αB̌k,r ,α B̂k,r ] G2 − G3 are the expected selling quantities per product. The
which are computed using Eq. (4) and shown as follows: maximum deviations from Gk = {630, 7, 4} and modifying
them to get a Type-2 fuzzy goal programming which can be
symmetrically handled where bk1 = bk2 = {10, 2, 2} and
bk1 = bk2 = {15, 3, 3}.
Using Eq. (7) we can obtain the values of the goals G1, G2
and G3 for every α-cut. The idea is then to minimize the
deviations from the goals through Eqs. (7), so we obtain four
crisp points that compose αz̃ and therefore z̃ as stated in Eq.
(19).
α-cut d11 d12 d21 d22 d31 d32
0.1 0 0 0 1.46 0 0
0.2 0 0 0 1.18 0 0
0.3 0 0 0 0 0 1.78
0.4 0 0 0 0 0 1.20
0.5 0 0 0 1.00 1.38 0
0.6 2.00 0 0 0 0 0
Figure 2: FGP 0.7 0 0 1.00 0 0 1.48
0.8 0 0 0 0 1.10 0
where αB̂k,l ,α B̌k,l are the left values of the cut for its UMF 0.9 0 0 0 0 1.68 0
and LMF respectively, and αB̌k,r ,α B̂k,r are the right values 1 0 0 1.13 0 0 0
of the cut for its LMF and UMF respectively. To do so, all
crisp boundaries of B̃k,r are computed as follows: Table 1: Optimal deviations for the left side UMF, LMF
α
B̂k,l = (bk − bk1 ) + α(bk − (bk − bk1 )), (11)
α As seen in Table 6, goal G2 was the only goal which ob-
B̌k,l = (bk − bk1 ) + α(bk − (bk − bk1 )), (12) tained its desired value on its left side while its right side has
α
B̂k,r = (bk + bk2 ) − α((bk + bk2 ) − bk ), (13) a linear behavior (see Table 6). There is a nonlinear behavior
α on all deviations from goals even when all goals were accom-
B̌k,r = (bk + bk2 ) − α((bk + bk2 ) − bk ), (14) plished, this is, there is no direct relationship between the
α-cut X1 X2 OF α-cut X1 X2 OF
0.1 6.66 2.20 1.46 0.1 5.09 5.80 3.71
0.2 6.58 2.40 1.18 0.2 5.18 5.60 3.43
0.3 5.60 4.38 1.78 0.3 5.26 5.40 3.14
0.4 5.80 4.00 1.20 0.4 5.35 5.20 2.85
0.5 7.00 1.63 2.38 0.5 5.44 5.00 2.56
0.6 6.20 3.20 2.00 0.6 5.53 4.80 2.28
0.7 5.40 4.88 2.48 0.7 5.61 4.60 1.99
0.8 6.60 2.50 1.10 0.8 5.70 4.40 1.70
0.9 6.80 2.13 1.68 0.9 5.79 4.20 1.41
1 5.88 4.00 1.13 1 5.88 4.00 1.13
Table 2: Optimal variables X1 , X2 for the left side UMF, Table 4: Optimal variables X1 , X2 for the left side UMF,
LMF LMF
α-cut d11 d12 d21 d22 d31 d32
of every goal B̃k that comes from the opinion of multiple
0.1 0 0 3.71 0 0 0
0.2 0 0 3.43 0 0 0
experts, so its optimal solution should be interpreted apart
0.3 0 0 3.14 0 0 0 from other α-cuts. A practical way to find a crisp solution is
0.4 0 0 2.85 0 0 0 by selecting an α-cut and then solve the problem keeping in
0.5 0 0 2.56 0 0 0 mind its results.
0.6 0 0 2.28 0 0 0
0.7 0 0 1.99 0 0 0 References
0.8 0 0 1.70 0 0 0 [Charnes and Cooper, 1961] A Charnes and WW Cooper.
0.9 0 0 1.41 0 0 0
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1 0 0 1.13 0 0 0
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