Navigator
- 模糊期望值模型
- Possibilistic expected mean-variance model
- Possibilistic expected mean-variance-skewness model
- 模糊机会约束规划
- 机会约束
- Maximax机会约束规划
- Minimax机会约束规划
- 定理
- 参考文献
模糊期望值模型
设
x
x
x为决策向量，
ξ
\xi
ξ为模糊向量，
f
(
x
,
ξ
)
f(x, \xi)
f(x,ξ)为目标函数，
g
j
(
x
,
ξ
)
g_j(x, \xi)
gj(x,ξ)表示约束函数，
j
=
1
,
2
,
…
,
p
j=1, 2, \dots, p
j=1,2,…,p，具有以下形式的模糊规划
max
f
(
x
,
ξ
)
s
.
t
.
g
j
(
x
,
ξ
)
≤
0
,
j
=
1
,
2
,
…
,
p
(1)
\max f(x, \xi)\\ s.t.\quad g_j(x, \xi)\leq 0, j=1, 2, \dots, p \tag{1}
maxf(x,ξ)s.t.gj(x,ξ)≤0,j=1,2,…,p(1)
但是模型
(
1
)
(1)
(1)并不是明确的规划函数，Liu给出了明确地模糊期望值模型.
考虑最大化期望效益的决策，可以建立如下单目标模糊期望值模型
max
E
[
f
(
x
,
ξ
)
]
s
.
t
.
E
[
g
j
(
x
,
ξ
)
]
≤
0
,
j
=
1
,
2
,
…
,
p
\max \mathbb{E}[f(x, \xi)]\\ s.t.\quad \mathbb{E}[g_j(x, \xi)]\leq 0, j=1, 2, \dots, p
maxE[f(x,ξ)]s.t.E[gj(x,ξ)]≤0,j=1,2,…,p
该问题的多目标规划形式如下
max
[
E
[
f
1
(
x
,
ξ
)
]
,
E
[
f
2
(
x
,
ξ
)
]
,
…
,
E
[
f
m
(
x
,
ξ
)
]
]
s
.
t
.
E
[
g
j
(
x
,
ξ
)
]
≤
0
,
j
=
1
,
2
,
…
,
p
\max [\mathbb{E}[f_1(x, \xi)], \mathbb{E}[f_2(x, \xi)], \dots, \mathbb{E}[f_m(x, \xi)]]\\ s.t.\quad \mathbb{E}[g_j(x, \xi)]\leq 0, j=1, 2, \dots, p
max[E[f1(x,ξ)],E[f2(x,ξ)],…,E[fm(x,ξ)]]s.t.E[gj(x,ξ)]≤0,j=1,2,…,p
为了平衡多个目标，建立建立目标规划模型
min
∑
j
=
1
l
P
j
∑
i
=
1
m
(
u
i
j
d
i
+
+
v
i
j
d
i
−
)
s
.
t
.
{
E
[
f
i
(
x
,
ξ
)
]
+
d
i
−
−
d
i
−
=
b
i
,
i
=
1
,
2
,
…
,
m
E
[
g
j
(
x
,
ξ
)
]
≤
0
,
j
=
1
,
2
,
…
,
p
d
i
+
,
d
i
−
≥
0
\min \sum_{j=1}^l P_j\sum_{i=1}^m(u_{ij}d_i^++v_{ij}d_i^-)\\ s.t. \begin{cases} \mathbb{E}[f_i(x, \xi)]+d_i^--d_i^-=b_i, \quad i=1,2, \dots, m\\ \mathbb{E}[g_j(x, \xi)]\leq 0,\quad j=1,2, \dots, p\\ d_i^+, d_i^-\geq 0 \end{cases}
minj=1∑lPji=1∑m(uijdi++vijdi−)s.t.⎩⎪⎨⎪⎧E[fi(x,ξ)]+di−−di−=bi,i=1,2,…,mE[gj(x,ξ)]≤0,j=1,2,…,pdi+,di−≥0
其中
P
j
P_j
Pj表示优先因子，表示各个目标的相对重要性，并且对于所有的
j
j
j，有
P
j
≫
P
j
+
1
P_j\gg P_{j+1}
Pj≫Pj+1，
u
i
j
u_{ij}
uij为对应优先级
j
j
j的第
i
i
i个目标的正偏差权重因子，
v
i
j
v_{ij}
vij相应的负偏差权重因子. 设置目标
i
i
i的目标值，得到关于目标的正负目标偏差为
d
i
+
=
[
E
[
f
(
x
,
ξ
)
]
−
b
i
]
∨
0
d_i^+=[\mathbb{E}[f(x, \xi)]-b_i]\vee 0
di+=[E[f(x,ξ)]−bi]∨0
以及
d
i
−
=
[
b
i
−
E
[
f
(
x
,
ξ
)
]
]
∨
0
d_i^-=[b_i-\mathbb{E}[f(x, \xi)]]\vee 0
di−=[bi−E[f(x,ξ)]]∨0
Possibilistic expected mean-variance model
As the description of the mean returns and risks of asset returns by the coherent trapezoidal fuzzy numbers, the possibilistic expected mean and variance for the coherent for the coherent trapezoidal fuzzy numbers are counterparts of the usual expected mean and variance of asset returns in Markowitzian mean-variance methodology. Therefore, following Markowitzian expected mean-variance methodology, in order to obtain an optimized portfolio the possibilistic expected mean can be maximized given the upper bound of the risk the investor can bear, i.e., the possibilistic variance of the portfolio. Specifically, the possibilistic expected mean-variance model for portfolio selection by the coherent trapezoidal fuzzy numbers can be structured as follows:
max E ( x 1 A 1 ⊕ x 2 A x ⋯ ⊕ x n A n ) s . t . { V ( x 1 A 1 ⊕ ⋯ ⊕ x n A n ) ≤ ν ∑ i = 1 n x i = 1 0 ≤ x i ≤ 1 , i = 1 , 2 , … , n \max \mathbb{E}(x_1A_1\oplus x_2A_x\dots \oplus x_nA_n)\\ s.t. \begin{cases} \mathbb{V}(x_1A_1\oplus \dots\oplus x_nA_n)\leq \nu\\ \sum_{i=1}^n x_i=1\\ 0\leq x_i\leq 1, i=1, 2, \dots, n \end{cases} maxE(x1A1⊕x2Ax⋯⊕xnAn)s.t.⎩⎪⎨⎪⎧V(x1A1⊕⋯⊕xnAn)≤ν∑i=1nxi=10≤xi≤1,i=1,2,…,n
Similarly, the possibilistic expected mean-variance model can restructured by minimizing the possibilistic variance given the lower bound of the possibilistic expected mean. In detail
min V ( x 1 A 1 ⊕ x 2 A 2 ⊕ ⋯ ⊕ x n A n ) s . t . { E ( x 1 A 1 ⊕ x 2 A 2 ⊕ ⋯ ⊕ x n A n ) ≥ α ∑ i = 1 n x i = 1 0 ≤ x i ≤ 1 , i = 1 , 2 , … , n \min V(x_1A_1\oplus x_2A_2\oplus\dots \oplus x_nA_n)\\ s.t. \begin{cases} \mathbb{E}(x_1A_1\oplus x_2A_2\oplus\dots \oplus x_nA_n)\geq \alpha\\ \sum_{i=1}^n x_i=1\\ 0\leq x_i\leq 1, i=1, 2, \dots, n \end{cases} minV(x1A1⊕x2A2⊕⋯⊕xnAn)s.t.⎩⎪⎨⎪⎧E(x1A1⊕x2A2⊕⋯⊕xnAn)≥α∑i=1nxi=10≤xi≤1,i=1,2,…,n
Possibilistic expected mean-variance-skewness model
The mean-variance-skewness model for portfolio selection was formulated and was investigated in the non-probabilistic framework.
max
S
(
x
1
A
1
⊕
x
2
A
2
⊕
⋯
⊕
x
n
A
n
)
s
.
t
.
{
E
(
x
1
A
1
⊕
x
2
A
2
⊕
⋯
⊕
x
n
A
n
)
≥
α
V
(
x
1
A
1
⊕
x
2
A
2
⊕
⋯
⊕
x
n
A
n
)
≤
ν
∑
i
=
1
n
x
i
=
1
0
≤
x
i
≤
1
,
i
=
1
,
2
,
…
,
n
\max S(x_1A_1\oplus x_2A_2\oplus\dots \oplus x_nA_n)\\ s.t. \begin{cases} \mathbb{E}(x_1A_1\oplus x_2A_2\oplus\dots \oplus x_nA_n)\geq \alpha\\ \mathbb{V}(x_1A_1\oplus x_2A_2\oplus\dots \oplus x_nA_n)\leq \nu\\ \sum_{i=1}^n x_i=1\\ 0\leq x_i\leq 1, i=1, 2, \dots, n \end{cases}
maxS(x1A1⊕x2A2⊕⋯⊕xnAn)s.t.⎩⎪⎪⎪⎨⎪⎪⎪⎧E(x1A1⊕x2A2⊕⋯⊕xnAn)≥αV(x1A1⊕x2A2⊕⋯⊕xnAn)≤ν∑i=1nxi=10≤xi≤1,i=1,2,…,n
模糊机会约束规划
机会约束
设
x
x
x为决策向量，
ξ
\xi
ξ为参数向量，
f
(
x
,
ξ
)
f(x, \xi)
f(x,ξ)为目标函数，
g
j
(
x
,
ξ
)
g_j(x, \xi)
gj(x,ξ)为约束函数，由于加入了模糊变量，可以希望约束以一定置信水平
α
\alpha
α成立，即有机会约束
C
r
{
(
g
j
(
x
,
ξ
)
)
≤
0
,
j
=
1
,
2
,
…
,
p
}
≥
α
Cr\{(g_j(x, \xi))\leq 0, j=1, 2, \dots, p\}\geq \alpha
Cr{(gj(x,ξ))≤0,j=1,2,…,p}≥α
Maximax机会约束规划
如果在一定置信水平成立的前提下，极大化目标函数的乐观值，可以有以下的模糊机会规划模型成立
max
f
ˉ
s
.
t
.
{
C
r
{
f
(
x
,
ξ
)
≥
f
ˉ
}
≥
β
C
r
{
g
i
(
x
,
ξ
)
≤
0
,
j
=
1
,
2
,
…
,
p
}
≥
α
\max \bar{f}\\ s.t. \begin{cases} Cr\{f(x, \xi)\geq \bar{f}\}\geq \beta\\ Cr\{g_i(x, \xi)\leq 0, j=1, 2, \dots, p\}\geq \alpha \end{cases}
maxfˉs.t.{Cr{f(x,ξ)≥fˉ}≥βCr{gi(x,ξ)≤0,j=1,2,…,p}≥α
其中
α
\alpha
α和
β
\beta
β为预先给定的置信水平，该模型可以等价于如下max-max形式，其中
f
ˉ
\bar{f}
fˉ为乐观值
max
x
max
f
ˉ
s
.
t
.
{
C
r
{
f
(
x
,
ξ
)
≥
f
ˉ
}
≥
β
C
r
{
g
j
(
x
,
ξ
)
≤
0
,
j
=
1
,
2
,
…
,
p
}
≥
α
\max_x\max \bar{f}\\ s.t. \begin{cases} Cr\{f(x, \xi)\geq \bar{f}\}\geq \beta\\ Cr\{g_j(x, \xi)\leq 0, j=1, 2, \dots, p\}\geq \alpha \end{cases}
xmaxmaxfˉs.t.{Cr{f(x,ξ)≥fˉ}≥βCr{gj(x,ξ)≤0,j=1,2,…,p}≥α
多目标决策模型为
max
x
[
max
f
ˉ
1
f
ˉ
1
,
max
f
ˉ
1
f
ˉ
2
,
…
,
max
f
ˉ
m
f
ˉ
m
]
s
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t
.
{
C
r
{
f
i
(
x
,
ξ
)
≥
f
ˉ
i
}
≥
β
ˉ
i
,
i
=
1
,
2
,
…
,
m
C
r
{
g
j
(
x
,
ξ
)
≤
0
}
≥
α
j
,
j
=
1
,
2
,
…
,
p
\max_x\bigg[ \max_{\bar{f}_1}\bar{f}_1, \max_{\bar{f}_1}\bar{f}_2, \dots,\max_{\bar{f}_m}\bar{f}_m\bigg]\\ s.t. \begin{cases} Cr\{f_i(x, \xi)\geq \bar{f}_i\}\geq \bar{\beta}_i, i=1, 2, \dots, m\\ Cr\{g_j(x, \xi)\leq 0\}\geq \alpha_j, j=1, 2, \dots, p \end{cases}
xmax[fˉ1maxfˉ1,fˉ1maxfˉ2,…,fˉmmaxfˉm]s.t.{Cr{fi(x,ξ)≥fˉi}≥βˉi,i=1,2,…,mCr{gj(x,ξ)≤0}≥αj,j=1,2,…,p
根据目标的优先结构和目标水平，还可以构建如下机会约束的目标规划
min
∑
j
=
1
l
P
j
∑
i
=
1
m
(
u
i
j
d
i
+
+
v
i
j
d
i
−
)
s
.
t
.
{
C
r
{
f
i
(
x
,
ξ
)
}
−
b
i
≤
d
i
+
}
≥
β
i
+
C
r
{
b
i
−
f
i
(
x
,
ξ
)
≤
d
i
−
}
≥
β
i
−
C
r
{
g
j
(
x
,
ξ
)
≤
0
}
≥
α
j
d
i
+
,
d
i
−
≥
0
\min \sum_{j=1}^l P_j\sum_{i=1}^m(u_{ij}d_i^++v_{ij}d_i^- )\\ s.t. \begin{cases} Cr\{f_i(x, \xi)\}-b_i\leq d_i^+\}\geq \beta_i^+\\ Cr\{b_i-f_i(x, \xi)\leq d_i^-\}\geq \beta_i^-\\ Cr\{g_j(x, \xi)\leq 0\}\geq \alpha_j\\ d_i^+, d_i^-\geq 0 \end{cases}
minj=1∑lPji=1∑m(uijdi++vijdi−)s.t.⎩⎪⎪⎪⎨⎪⎪⎪⎧Cr{fi(x,ξ)}−bi≤di+}≥βi+Cr{bi−fi(x,ξ)≤di−}≥βi−Cr{gj(x,ξ)≤0}≥αjdi+,di−≥0
在模糊向量退化为清晰向量时，置信水平约束条件退化为
d
i
+
=
[
f
i
(
x
,
ξ
)
−
b
i
]
∨
0
d
i
−
=
[
b
i
−
f
i
(
x
,
ξ
)
]
∨
0
d_i^+=[f_i(x, \xi)-b_i]\vee 0\\ d_i^-=[b_i-f_i(x, \xi)]\vee 0
di+=[fi(x,ξ)−bi]∨0di−=[bi−fi(x,ξ)]∨0
Minimax机会约束规划
和Robust optimization中的worst case思想一致，可以考虑极大化目标函数的悲观值
max
x
min
f
ˉ
s
.
t
.
{
C
r
{
f
(
x
,
ξ
)
≤
f
ˉ
}
≥
β
C
r
{
g
j
(
x
,
ξ
)
≤
0
,
j
=
1
,
2
,
…
,
p
}
≥
α
\max_x\min \bar{f}\\ s.t. \begin{cases} Cr\{f(x, \xi)\leq \bar{f}\}\geq \beta\\ Cr\{g_j(x, \xi)\leq 0, j=1, 2, \dots, p\}\geq \alpha \end{cases}
xmaxminfˉs.t.{Cr{f(x,ξ)≤fˉ}≥βCr{gj(x,ξ)≤0,j=1,2,…,p}≥α
同样可以推导出多目标的情况
max
x
[
min
f
ˉ
1
,
min
f
ˉ
2
,
…
,
min
f
ˉ
m
]
s
.
t
.
{
C
r
{
f
i
(
x
,
ξ
)
≤
f
ˉ
i
}
≥
β
i
C
r
{
g
j
(
x
,
ξ
)
≤
0
,
j
=
1
,
2
,
…
,
p
}
≥
α
j
\max_x\bigg[\min \bar{f}_1, \min \bar{f}_2, \dots, \min\bar{f}_m\bigg]\\ s.t. \begin{cases} Cr\{f_i(x, \xi)\leq \bar{f}_i\}\geq \beta_i\\ Cr\{g_j(x, \xi)\leq 0, j=1, 2, \dots, p\}\geq \alpha_j \end{cases}
xmax[minfˉ1,minfˉ2,…,minfˉm]s.t.{Cr{fi(x,ξ)≤fˉi}≥βiCr{gj(x,ξ)≤0,j=1,2,…,p}≥αj
定理
设
f
P
,
f
N
,
f
C
f_P, f_N, f_C
fP,fN,fC分别是在Pos，Nec和Cr测度下的目标最优值，则关系
f
N
≤
f
C
≤
f
P
f_N\leq f_C\leq f_P
fN≤fC≤fP
成立.
根据定义可以推导出
f
C
=
max
x
∈
S
C
max
f
ˉ
{
f
ˉ
∣
C
r
{
f
(
x
,
ξ
)
≥
f
ˉ
}
≥
β
}
≤
max
x
∈
S
C
max
f
ˉ
{
f
ˉ
∣
P
o
s
{
f
(
x
,
ξ
)
≥
f
ˉ
}
≥
β
}
≤
max
x
∈
S
P
max
f
ˉ
{
f
ˉ
∣
P
o
s
{
f
(
x
,
ξ
)
}
≥
f
ˉ
}
≥
β
}
=
f
P
\begin{aligned} f_C&=\max_{x\in S_C}\max_{\bar{f}}\{\bar{f}\mid Cr\{f(x, \xi)\geq \bar{f}\}\geq \beta\}\\ &\leq \max_{x\in S_C}\max_{\bar{f}}\{\bar{f}\mid Pos\{f(x, \xi)\geq \bar{f}\}\geq \beta \}\\ &\leq \max_{x\in S_P}\max_{\bar{f}}\{\bar{f}\mid Pos\{f(x, \xi)\}\geq \bar{f}\}\geq \beta\}\\ &=f_P \end{aligned}
fC=x∈SCmaxfˉmax{fˉ∣Cr{f(x,ξ)≥fˉ}≥β}≤x∈SCmaxfˉmax{fˉ∣Pos{f(x,ξ)≥fˉ}≥β}≤x∈SPmaxfˉmax{fˉ∣Pos{f(x,ξ)}≥fˉ}≥β}=fP
根据Hurwicz准则，对maximax和minimax模型分别赋予
λ
\lambda
λ和
1
−
λ
1-\lambda
1−λ的权重，其中
λ
\lambda
λ表示悲观程度
max
λ
f
min
+
(
1
−
λ
)
f
max
s
.
t
.
P
o
s
/
N
e
c
/
C
r
{
g
j
(
x
,
ξ
)
≤
0
,
j
=
1
,
2
,
…
,
p
}
≥
α
\max \lambda f_{\min}+(1-\lambda)f_{\max}\\ s.t.\quad Pos/Nec/Cr\{g_j(x, \xi)\leq 0, j=1, 2, \dots, p\}\geq \alpha
maxλfmin+(1−λ)fmaxs.t.Pos/Nec/Cr{gj(x,ξ)≤0,j=1,2,…,p}≥α
参考文献
不确定规划 清华大学出版社
Portfolio selection with coherent investor’s expectations under uncertainty
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