模糊期望值模型

設$x$為決策向量，$\xi$為模糊向量，$f(x,\xi )$為目標函式，$g_j(x,\xi )$表示約束函式，$j=1,2,\dots ,p$，具有以下形式的模糊規劃
$$\begin{gathered} \max f(x,\xi ) \\ s.t.g_j(x,\xi )\le 0,j=1,2,\dots ,p\text{\hspace{1em}(1)} \end{gathered}$$
但是模型$(1)$並不是明確的規劃函式，Liu給出了明確地模糊期望值模型.
考慮最大化期望效益的決策，可以建立如下單目標模糊期望值模型
$$\begin{gathered} \max \mathbb{E}[f(x,\xi )] \\ s.t.\mathbb{E}[g_j(x,\xi )]\le 0,j=1,2,\dots ,p \end{gathered}$$
該問題的多目標規劃形式如下
$$\begin{gathered} \max [\mathbb{E}[f_1(x,\xi )],\mathbb{E}[f_2(x,\xi )],\dots ,\mathbb{E}[f_m(x,\xi )]] \\ s.t.\mathbb{E}[g_j(x,\xi )]\le 0,j=1,2,\dots ,p \end{gathered}$$
為了平衡多個目標，建立建立目標規劃模型
$$\begin{gathered} \min \sum _{j=1}^l{P}_j\sum _{i=1}^m(u_{ij}{d}_i^{+}+v_{ij}{d}_i^{-}) \\ s.t.\{\begin{array}{l}\mathbb{E}[f_i(x,\xi )]+d_i^{-}-d_i^{-}=b_i,i=1,2,\dots ,m\\ \mathbb{E}[g_j(x,\xi )]\le 0,j=1,2,\dots ,p\\ d_i^{+},d_i^{-}\ge 0\end{array} \end{gathered}$$
其中$P_j$表示優先因子，表示各個目標的相對重要性，並且對於所有的$j$，有$P_j\gg P_{j+1}$，$u_{ij}$為對應優先順序$j$的第$i$個目標的正偏差權重因子，$v_{ij}$相應的負偏差權重因子. 設定目標$i$的目標值，得到關於目標的正負目標偏差為
$$d_i^{+}=[\mathbb{E}[f(x,\xi )]-b_i]\vee 0$$
以及
$$d_i^{-}=[b_i-\mathbb{E}[f(x,\xi )]]\vee 0$$

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:

$$\begin{gathered} \max \mathbb{E}(x_1{A}_1\oplus x_2{A}_x\cdots \oplus x_n{A}_n) \\ s.t.\{\begin{array}{l}\mathbb{V}(x_1{A}_1\oplus \cdots \oplus x_n{A}_n)\le \nu \\ {\sum }_{i=1}^n{x}_i=1\\ 0\le x_i\le 1,i=1,2,\dots ,n\end{array} \end{gathered}$$

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

$$\begin{gathered} \min V(x_1{A}_1\oplus x_2{A}_2\oplus \cdots \oplus x_n{A}_n) \\ s.t.\{\begin{array}{l}\mathbb{E}(x_1{A}_1\oplus x_2{A}_2\oplus \cdots \oplus x_n{A}_n)\ge \alpha \\ {\sum }_{i=1}^n{x}_i=1\\ 0\le x_i\le 1,i=1,2,\dots ,n\end{array} \end{gathered}$$

Possibilistic expected mean-variance-skewness model

The mean-variance-skewness model for portfolio selection was formulated and was investigated in the non-probabilistic framework.
$$\begin{gathered} \max S(x_1{A}_1\oplus x_2{A}_2\oplus \cdots \oplus x_n{A}_n) \\ s.t.\{\begin{array}{l}\mathbb{E}(x_1{A}_1\oplus x_2{A}_2\oplus \cdots \oplus x_n{A}_n)\ge \alpha \\ \mathbb{V}(x_1{A}_1\oplus x_2{A}_2\oplus \cdots \oplus x_n{A}_n)\le \nu \\ {\sum }_{i=1}^n{x}_i=1\\ 0\le x_i\le 1,i=1,2,\dots ,n\end{array} \end{gathered}$$

模糊機會約束規劃

機會約束

設$x$為決策向量，$\xi$為引數向量，$f(x,\xi )$為目標函式，$g_j(x,\xi )$為約束函式，由於加入了模糊變數，可以希望約束以一定置信水平$\alpha$成立，即有機會約束
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機會約束規劃

如果在一定置信水平成立的前提下，極大化目標函式的樂觀值，可以有以下的模糊機會規劃模型成立
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} maxfˉ​s.t.{Cr{f(x,ξ)≥fˉ​}≥βCr{gi​(x,ξ)≤0,j=1,2,…,p}≥α​
其中$\alpha$和$\beta$為預先給定的置信水平，該模型可以等價於如下max-max形式，其中$\bar{f}$為樂觀值
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} xmax​maxfˉ​s.t.{Cr{f(x,ξ)≥fˉ​}≥βCr{gj​(x,ξ)≤0,j=1,2,…,p}≥α​
多目標決策模型為
max ⁡ x [ max ⁡ f ˉ 1 f ˉ 1 , max ⁡ f ˉ 1 f ˉ 2 , … , max ⁡ f ˉ m f ˉ m ] s . 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​[fˉ​1​max​fˉ​1​,fˉ​1​max​fˉ​2​,…,fˉ​m​max​fˉ​m​]s.t.{Cr{fi​(x,ξ)≥fˉ​i​}≥βˉ​i​,i=1,2,…,mCr{gj​(x,ξ)≤0}≥αj​,j=1,2,…,p​
根據目標的優先結構和目標水平，還可以構建如下機會約束的目標規劃
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∑l​Pj​i=1∑m​(uij​di+​+vij​di−​)s.t.⎩⎪⎪⎪⎨⎪⎪⎪⎧​Cr{fi​(x,ξ)}−bi​≤di+​}≥βi+​Cr{bi​−fi​(x,ξ)≤di−​}≥βi−​Cr{gj​(x,ξ)≤0}≥αj​di+​,di−​≥0​
在模糊向量退化為清晰向量時，置信水平約束條件退化為
$$\begin{gathered} d_i^{+}=[f_i(x,\xi )-b_i]\vee 0 \\ {d}_i^{-}=[b_i-f_i(x,\xi )]\vee 0 \end{gathered}$$

Minimax機會約束規劃

和Robust optimization中的worst case思想一致，可以考慮極大化目標函式的悲觀值
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} xmax​minfˉ​s.t.{Cr{f(x,ξ)≤fˉ​}≥βCr{gj​(x,ξ)≤0,j=1,2,…,p}≥α​
同樣可以推匯出多目標的情況
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​[minfˉ​1​,minfˉ​2​,…,minfˉ​m​]s.t.{Cr{fi​(x,ξ)≤fˉ​i​}≥βi​Cr{gj​(x,ξ)≤0,j=1,2,…,p}≥αj​​

定理

設$f_P,f_N,f_C$分別是在Pos，Nec和Cr測度下的目標最優值，則關係
$$f_N\le f_C\le f_P$$
成立.
根據定義可以推匯出
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∈SC​max​fˉ​max​{fˉ​∣Cr{f(x,ξ)≥fˉ​}≥β}≤x∈SC​max​fˉ​max​{fˉ​∣Pos{f(x,ξ)≥fˉ​}≥β}≤x∈SP​max​fˉ​max​{fˉ​∣Pos{f(x,ξ)}≥fˉ​}≥β}=fP​​
根據Hurwicz準則，對maximax和minimax模型分別賦予$\lambda$和$1-\lambda$的權重，其中$\lambda$表示悲觀程度
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)fmax​s.t.Pos/Nec/Cr{gj​(x,ξ)≤0,j=1,2,…,p}≥α

參考文獻

不確定規劃 清華大學出版社
Portfolio selection with coherent investor’s expectations under uncertainty