In many applications, the subjective assessments provided by DM(s) are usually conceptually vague, with uncertainty that is frequently represented in linguistic forms.
To help people easily express their subjective assessments, the linguistic variables [163]
are used to linguistically express requirements. Assume that the fuzzy targets linguis-tically specified by the DM have the canonical form, and π(t) is the membership de-gree/possibility distribution of fuzzy target T.
For notational convenience, we also designate an evaluation attribute by X, and an arbitrary specific level of that evaluation attribute by x. We also restrict the variable x to a bounded domain D = [Xmin, Xmax]. Suppose that a DM has to rank several possible decisions A = {A1,· · · , Am,· · ·, AM}, where Am represent the alternatives (or acts) available to a DM, one of which must be selected. Assume for simplicity that the set Aof consequences is finite and completely ordered by a preference relation. By a fuzzy target we mean a possibility variable T over the attribute domain D, by a possibility distribution T : D → [0,1]. We also assume further that T is a piecewise continuous function having a bounded support and
DT(t)dt > 0.
Given a fuzzy targetT, letπ(t) be the possibility distribution function, andp(t) be its associated probability distribution function. According to the proportional transformation method, we can obtain the induced probability distribution as follows:
p(t) = π(t)
tπ(t)dt. (4.8)
In the following, we shall extend the two probabilistic target-oriented decision models into the fuzzy target-oriented decision analysis case.
4.4.1 Cumulative Distribution Function Based Method
Probabilistic target-oriented decision model suggests using the following ranking of alternatives be obtained by using the value function defined by
Val(Am) =
x
Pr(xT)pAm(x)
wherepAm is the probability distribution for the random consequence Xm associated with an act Am,T is an uncertain target having a random distribution onD, Pr(xT) is the probability of meeting the uncertain targetT and T is stochastically independent ofXm. As we mentioned in Chapter 3, there are three type of target preferences: benefit target, cost target, and equal/range level target. Based on probabilistic target-oriented decision model and the induced pdf of fuzzy target T, we can obtain the probability of meeting target as follows:
Pr(xT) = Xmax
Xmin u(x, t)π(t)dt Xmax
Xmin π(t)dt . (4.9)
In case of benefit target preference, we can obtain the target achievement function as Pr(xT) = Pr(x≥T)
= x
Xminπ(t)dt Xmax
Xmin π(t)dt
(4.10)
In case of cost target preference, we can obtain the target achievement function as Pr(xT) = Pr(x≤T)
= Xmax
x π(t)dt Xmax
Xmin π(t)dt. (4.11)
When the DM has equal/range level target preference, as the target-oriented decision model views the mode value of the pdf as a reflection point, there will be added loss of value for missing the reference point on the low side, or added loss for exceeding the reference point. We first consider the situation that the pdf is unimodal, the mode value is denoted by Tm. Then the fuzzy target-oriented decision model with respect to equal target preference can be defined as
Pr(x∼=T) =
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
Rx
Xminπ(t)dt RTm
Xminπ(t)dt, if x < Tm; 1, else ifx=Tm;
RXmax
x π(t)dt RXmax
Tm π(t)dt, otherwise.
(4.12)
Generally speaking, the reference point Tm may have a interval range, such thatTm ≡ [Tml, Tmu]. In this case, the induced value function is defined as follows:
Pr(x∼=T) =
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
Rx
Xminπ(t)dt RTml
Xminπ(t)dt, if x < Tml;
1, else if x∈[Tml, Tmu];
RXmax
x π(t)dt RXmax
Tmu π(t)dt, if x > Tmu.
(4.13)
4.4.2 Level Set Based Approach
In Chapter 3, a level set based approach has been proposed to deal with the target-oriented decision making with probabilistic uncertainty. The first step is to converse the fuzzy target T into probability distribution such that
p(t) = π(t)
xπ(t).
Let T be the induced probabilistic target having a random pdf over the bounded do-mainD= [Xmin, Xmax]. Letσbe any given probability level, where 0 ≤σ≤supT (supT denotes the support of the pdf of uncertain target), Tσ consists of all the elements whose probabilities are greater than or equal toσ such that
Tσ ={t∈D|p(t)≥σ}, (4.14)
Tσ is called the σ-level set of random target T. It should be noted that target-oriented decision analysis assumes that the uncertain target has a unimodal pdf, thus we can express as Tσ = [Tσl, Tσr], where Tσl and Tσr are the left and right bound of level cut, respectively.
Based on the distribution function of level sets of pdf provided before, similar but different from Garcia et al. [45], we define the following function:
Pr(xT) =
supT 0
u(x, Tσ)Tσdσ, (4.15) where u(x, Tσ) indicates the degree that the target achievement in the level set Tσ, supT = R 1
tπ(t)dt denotes the support of the pdf of uncertain target, and u(x, Tσ) ∈[0,1], supu(x, Tσ) = 1.
Considering different target preferences, we further define
Pr(xT) =
⎧⎪
⎨
⎪⎩
Pr(x≥T) =supT
0 u(x≥Tσ)Tσdσ, benefit target preference;
Pr(x≤T) =supT
0 u(x≤Tσ)Tσdσ, cost target preference;
Pr(x∼=T) =supT
0 u(x∼=Tσ)Tσdσ, equal/range target preference.
(4.16) The second step is to calculate the target achievement with respect to different target preferences based on the induced probability distribution function.
Benefit target preference
If the DM has a monotonically increasing target preference, for an interval Tσ = [Tσl, Tσr], to ensure that u(x, Tσ)∈[0,1] and supu(x, Tσ) = 1, we define
u(x≥Tσ) = Tσr
Tσl u(x, t)p(t)dt Tσr
Tσl p(t)dt (4.17)
As target-oriented model assumes that there are only two levels of utility (1 or 0), thus we define
u(x, t) =
1, if x≥t;
0, otherwise.
whereu(x, t) denotes whether the attribute level achieves target level or not. Then we can obtain u(x≥Tσ) as follows:
u(x≥Tσ) =
⎧⎪
⎪⎨
⎪⎪
⎩
0,R if x < Tσl;
T lσx p(t)dt RT rσ
T lσ p(t)dt, if Tσl ≤x≤Tσr; 1, if x > Tσr.
(4.18)
By substituting Eq. (4.18) into the general representation of level set based target-oriented decision model Eq. (4.16), we can obtain the probability of meeting uncer-tain target T.
Cost target preference
Similarly, in case of cost target preference, we define u(x, t) =
1, if x≤t;
0, otherwise.
and then we can obtain u(x≤Tσ) as follows:
u(x≤Tσ) =
⎧⎪
⎪⎨
⎪⎪
⎩
1, if x < Tσl;
RT rσ
x p(t)dt RT rσ
T lσ p(t)dt, if Tσl ≤x≤Tσr; 0, if x > Tσr.
(4.19)
It is clear that u(x≤Tσ) = 1−u(x≥Tσ), thus we obtain Pr(x≤T) =
supT 0
u(x≤Tσ)Tσdσ
=
supT 0
(1−u(x≥Tσ))Tσdσ
=
supT 0
Tσdσ− supT
0
u(x≥Tσ)Tσdσ
= 1−Pr(x≥T)
(4.20)
Equal/range target preference
In case of non-monotonic target preference, there exists an “ideal” level. Recall that target-oriented decision analysis views the modal value Tm of the pdf as reference point (reflection point), then there will be added loss of value for missing the refer-ence point on the low side, or added loss for exceeding the referrefer-ence point. In other words, when x=Tm the probability of meeting target should be equivalent to one;
when x < Tm it can be viewed as pseudo benefit attribute; and when x > Tm it can be viewed as pseudo cost attribute. Due to this observation, we can define the following function:
1. When x < Tm,
u(x∼=Tσ) =
⎧⎨
⎩
0,R if x < Tσl;
x T lσp(t)dt RTm
T lσ p(t)dt, otherwise. (4.21) 2. When x=Tm,
u(x∼=Tσ) = x
Tmp(t)dt Tm
Tm p(t)dt = 1 (4.22)
3. When x > Tm,
u(x∼=Tσ) =
⎧⎨
⎩
0, if x > Tσr;
RT rσ
x p(t)dt RT rσ
Tmp(t)dt, otherwise. (4.23) It should be noted that if the mode value Tm is an interval range, such that Tm ≡ [Tml, Tmu], then we can define u(x∼= Tσ) = 1 if Tml ≤ x ≤ Tmu. Typical examples of this case are the trapezoidal distributions.
In the following subsection, we shall use four commonly used fuzzy targets in decision making to illustrate the proposed model.