信息集成算子是多属性决策研究的重要内容之一.当前, 信息集成算子主要有(加权)算术平均算子(WA)^[1]、(加权)几何平均算子(WG)^[2]、有序加权平均算子(OWA)^[3]、有序加权几何平均算子(OWG)^[4]、有序加权调和平均算子(OWH)^[5]、广义加权有序平均算子(GOWA)^[6]、拟有序加权平均算子(QOWA)^[7]等.最近, 在QOWA算子的启发下, 刘卫锋等^[8]将拟函数与有序加权几何算子相结合, 定义了拟有序加权几何算子(QOWG), 并将其推广至毕达哥拉斯模糊决策环境.QOWG算子的提出丰富了集成算子的类型, 发展了集成算子理论.

由于实际决策中经常碰到区间数决策信息的情况, 许多学者开始研究区间数决策信息集成, 并取得了一系列研究成果.其中, XU等^{[9 — 11]}提出了UOWA算子、UOWG算子、区间数幂均算子; MERIGÓ等研究了UIOWAWA算子^[12]及区间诱导QOWA集成算子^[13]; RAN等^[14]将优先加权平均算子推广到区间数, 定义了UPWA算子、UPWG算子及UPWHA算子; YAGER等^{[15 — 16]}研究了COWA算子以及COWG算子, XU^[17]将COWA算子应用于求解区间模糊偏好关系的排序向量; 徐泽水^[18]定义了WCOWA算子、OWCOWA算子及CCOWA算子; 龚艳冰等^[19]提出了模糊COWA算子; 汪新凡^[20]定义了WCOWG算子、OWCOWG算子及CCOWG算子; 丁德臣等^[21]定义了模糊COWG算子; WU等^[22]研究了ICOWG算子; 陈华友等^[23]探讨了COWH算子; ZHOU等^[24]提出了CGOWA算子及其扩展; LIU等^[25]定义了CQOWA算子.分析发现, 上述集成算子大致可分为2类:一类是以XU为代表的将集成算子直接由实数推广至区间数, 如文献[9-14].但此类算子在集成过程中涉及区间数的各种运算, 利用可能度^[9]等方法实现区间数的排序, 而区间数运算往往易导致不确定性的增加, 使得到的结果出现较大误差甚至失真.另一类是以YAGER和XU为代表, 主要通过引入反映决策者决策态度的态度参数将区间数转化为实数, 然后进行集成, 如文献[15-25].该类算子避免了区间数运算和排序引发的系列问题.

由于目前QOWG算子仅适用于精确数信息的集成, 因而有必要将其推广至区间数决策环境, 定义连续区间QOWG算子, 扩大QOWG算子的决策应用范围.借助生成函数, 连续区间QOWG算子可将多种信息集成为算子综合在一个函数表达式之中, 故而在决策中, 决策者只需通过控制生成函数即可实现决策结果的调整, 从而令不同的生成函数对应于不同的集成算子; 同时, 连续区间QOWG算子中态度参数的选择也给决策者带来了很大的主动性.因此, 研究连续区间QOWG算子, 对于发展QOWG集成算子以及区间数集成算子均具有重要的理论和现实意义.

根据以上分析, 笔者尝试将QOWG算子推广至连续区间的数决策环境.首先, 定义连续QOWG算子(continuous QOWG operator, CQOWG), 讨论其性质和特殊形式.其次, 定义CQOWG算子的orness测度, 探讨orness测度的性质.然后, 拓展CQOWG算子, 定义了加权连续QOWG算子(WCQOWG)、有序加权连续QOWG算子(OWCQOWG)以及组合连续QOWG算子(CCQOWG), 并探讨了它们的性质.最后, 提出基于CQOWA算子的群决策方法, 并通过决策实例说明其可行性与有效性.

1 相关概念

定义1^[8]   设a_i∈R⁺(i=1, 2, …, n)为一组待集成数据, 其中R⁺={x|x>0}, 若函数

$ {\rm{QOWG}}: = {\left( {{R^ + }} \right)^n} \to {R^ + }, $

$ {\rm{QOWG}}\left( {{a_1},{a_2}, \cdots ,{a_n}} \right) = {f^{ - 1}}\left( {\prod\limits_{j = 1}^n {{{\left( {f\left( {{b_j}} \right)} \right)}^{{w_j}}}} } \right), $

则称QOWG为拟有序加权几何算子, 简称QOWG算子, 其中b_j表示a_i(i=1, 2, …, n)中第j大的数, f为严格单调连续函数, 而(w₁, w₂, …, w_n)为与QOWG算子相关联的权重向量, w_j≥0且$\sum\limits_{j = 1}^n {{w_j}} = 1$.

定义2^[15]   若函数F:Ω→R⁺满足

$ {F_Q}\left( {\left[ {a,b} \right]} \right) = \int_0^1 {\frac{{{\rm{d}}Q\left( y \right)}}{{{\rm{d}}y}}\left( {b - y\left( {b - a} \right)} \right){\rm{d}}y} , $

则称F为COWA算子, 其中[a, b]∈Ω, Ω={[a, b]| 0 < a≤b}, F与Q有关, 称Q为基本单位区间单调函数(BUM), 满足Q:[0, 1]→[0, 1], 且Q(0)=0, Q(1)=1.

定理1^[15]   设$\mu = \int_0^1 {Q\left( y \right){\rm{d}}y} $, 则有

$ {F_Q}\left( {\left[ {a,b} \right]} \right) = \mu b + \left( {1 - \mu } \right)a, $

其中, μ为BUM函数Q的态度参数.

定义3^[16]   若函数G:Ω→R⁺满足

$ {G_Q}\left( {\left[ {a,b} \right]} \right) = b\left( {\frac{a}{b}} \right)\int_0^1 {\frac{{{\rm{d}}Q\left( y \right)}}{{{\rm{d}}y}}y{\rm{d}}y} , $

则称G为COWG算子, 其中[a, b]∈Ω, G与Q有关, Q为BUM函数.

定理2^[16]   设$\mu = \int_0^1 {Q\left( y \right){\rm{d}}y} $为BUM函数Q的态度参数, 则有G_Q([a, b])=b^μa^1-μ.

定义4^[23]   若函数H:Ω→R⁺满足

$ {H_Q}\left( {\left[ {a,b} \right]} \right) = \frac{1}{{\int_0^1 {\frac{{{\rm{d}}Q\left( y \right)}}{{{\rm{d}}y}}\left( {\frac{1}{b} - y\left( {\frac{1}{a} - \frac{1}{b}} \right)} \right){\rm{d}}y} }}, $

则称H为COWH算子, 其中[a, b]∈Ω, H与Q有关, Q为BUM函数.

定理3^[23]   设$\mu = \int_0^1 {Q\left( y \right){\rm{d}}y} $为BUM函数Q的态度参数, 则有H_Q([a, b])= ${H_Q}\left( {\left[ {a,b} \right]} \right) = 1/\left[ {\frac{\mu }{b} + \frac{{1 - \mu }}{a}} \right]$.

定义5^[24]   若函数g:Ω→R⁺满足

$ {g_Q}\left( {\left[ {a,b} \right]} \right) = {\left( {\int_0^1 {\frac{{{\rm{d}}Q\left( y \right)}}{{{\rm{d}}y}}\left( {{b^\lambda } - y\left( {{b^\lambda } - {a^\lambda }} \right)} \right){\rm{d}}y} } \right)^{1/\lambda }}{\rm{d}}y, $

则称函数g为连续区间广义OWA算子, 简称CGOWA算子, 其中[a, b]∈Ω, g与Q有关, Q为BUM函数, 参数λ∈R-{0}.

定理4^[24]   设$\mu = \int_0^1 {Q\left( y \right){\rm{d}}y} $, 则有g_Q([a, b])=(μb^λ+(1-μ)a^λ)^1/λ, 其中, μ为BUM函数Q的态度参数.

定义6^[25]   若函数φ:Ω→R⁺满足

$ {\varphi _{Q,f}}\left( {\left[ {a,b} \right]} \right) = {f^{ - 1}}\left\{ {\int_0^1 {\frac{{{\rm{d}}Q\left( y \right)}}{{{\rm{d}}y}}\left( {f\left( b \right) - y\left( {f\left( b \right) - f\left( b \right)} \right)} \right){\rm{d}}y} } \right\}, $

则称函数φ为连续区间拟OWA算子, 简称CQOWA算子, 其中[a, b]∈Ω, φ与Q有关, Q为与函数φ相关联的BUM函数, f为[a, b]上严格单调的连续函数, 称为φ_{Q, f}([a, b])的导出函数.

定理5^[25]  设$\mu = \int_0^1 {Q\left( y \right){\rm{d}}y} $, 则有

$ {\varphi _{Q,f}}\left( {\left[ {a,b} \right]} \right) = {f^{ - 1}}\left[ {f\left( a \right) - \mu \left( {f\left( b \right) - f\left( a \right)} \right)} \right], $

其中μ为BUM函数Q的态度参数.

2 连续拟有序加权几何算子

定义7   设[a, b]∈Ω, 若函数φ:Ω→R⁺满足

$ {\varphi _{Q,f}}\left( {\left[ {a,b} \right]} \right) = {f^{ - 1}}\left\{ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{\int_0^1 {\frac{{{\rm{d}}Q\left( y \right)}}{{{\rm{d}}y}}y{\rm{d}}y} }}} \right\}, $

则称函数φ为连续区间拟OWG算子, 简称CQOWG算子, 其中, Q为与函数φ相关联的BUM函数, f为[a, b]上严格单调的连续函数, 称为φ_{Q, f}([a, b])的导出函数.

下面从定积分角度说明定义7中公式的由来.

设[a, b]∈Ω, f为[a, b]上的严格单调连续函数, Q(y)为BUM函数, 则由BUM函数可得到QOWG算子的一组加权向量

$ {\mathit{\boldsymbol{w}}_j} = Q\left( {\frac{j}{n}} \right) - Q\left( {\frac{{j - 1}}{n}} \right),j = 1,2, \cdots ,n, $

满足w_j∈[0, 1], j=1, 2, …, n且$\sum\limits_{j = 1}^n {{w_j}} = 1$.

下面构造一组离散的数据集合逼近连续区间数[a, b].

令$\delta = {\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)^{1/n}},{且}\;{q_j} = {f^{ - 1}}\left[ {f\left( b \right){\delta ^j}} \right] = {f^{ - 1}}\left[ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{j/n}}} \right]$, j=0, 1, 2, …, n.当j=0时, q_j=b; 当j=n时, q_j=a.当f(x)为严格单调连续递增函数时, f^-1(x)也是严格单调连续递增函数, 故f(a)≤f(b), 即有$\frac{{f\left( a \right)}}{{f\left( b \right)}} \le 1 $, 于是${\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)^{\frac{j}{n}}} \ge {\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)^{\frac{{j + 1}}{n}}}$, 故

$ f\left[ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{\frac{j}{n}}}} \right] \ge f\left[ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{\frac{{j + 1}}{n}}}} \right], $

即q_j≥q_j+1, 于是得到q₀≥q₁≥q₂≥…≥q_n.当f(x)严格单调连续递减时, 同理也可得到q₀≥q₁≥q₂≥…≥q_n.

根据QOWG算子, 可以得到

$ \begin{array}{l} {\varphi _{Q,f}}\left( {\left[ {a,b} \right]} \right) \approx {\rm{QOWG}}\left( {{q_1},{q_2}, \cdots ,{q_n}} \right) = \\ \;\;\;\;\;\;{f^{ - 1}}\left\{ {\prod\limits_{j = 1}^n {{{\left( {f\left( {{q_j}} \right)} \right)}^{{w_j}}}} } \right\} = \\ \;\;\;\;\;\;{f^{ - 1}}\left\{ {\prod\limits_{j = 1}^n {{{\left( {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{j/n}}} \right)}^{Q\left( {\frac{j}{n}} \right) - Q\left( {\frac{{j - 1}}{n}} \right)}}} } \right\} = \\ \;\;\;\;\;\;{f^{ - 1}}\left\{ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{\sum\limits_{j = 1}^n {\frac{j}{n}\left( {Q\left( {\frac{j}{n}} \right) - Q\left( {\frac{{j - 1}}{n}} \right)} \right)} }}} \right\}. \end{array} $

令Δy=1/n, 有

$ \begin{array}{l} {\varphi _{Q,f}}\left( {\left[ {a,b} \right]} \right) \approx \\ \;\;\;\;\;\;{f^{ - 1}}\left\{ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{\sum\limits_{j = 1}^n {j\Delta y\left( {Q\left( {j\Delta y} \right) - Q\left( {j\Delta y - \Delta y} \right)} \right)} }}} \right\} = \\ \;\;\;\;\;\;{f^{ - 1}}\left\{ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{\sum\limits_{j = 1}^n {j\Delta y\left( {\frac{{Q\left( {j\Delta y} \right) - Q\left( {j\Delta y - \Delta y} \right)}}{{\Delta y}}} \right)\Delta y} }}} \right\}. \end{array} $

令y=jΔy, 则当j从0取到n时, 有y∈[0, 1], 对上式两端取极限, 令n→+∞, 得到

$ {\varphi _{Q,f}}\left( {\left[ {a,b} \right]} \right) = {f^{ - 1}}\left\{ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{\int_0^1 {\frac{{{\rm{d}}Q\left( y \right)}}{{{\rm{d}}y}}y{\rm{d}}y} }}} \right\}. $

定理6   设$\mu = \int_0^1 {Q\left( y \right){\rm{d}}y} $为BUM函数Q的态度参数, 则有φ_{Q, f}([a, b])=f^-1[f(a)^1-μf(b)^μ].

证明

$ \begin{array}{l} {\varphi _{Q,f}}\left( {\left[ {a,b} \right]} \right) = {f^{ - 1}}\left\{ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{\int_0^1 {\frac{{{\rm{d}}Q\left( y \right)}}{{{\rm{d}}y}}y{\rm{d}}y} }}} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{f^{ - 1}}\left\{ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{\int_0^1 {y{\rm{d}}Q\left( y \right)} }}} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{f^{ - 1}}\left\{ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{1 - \int_0^1 {Q\left( y \right){\rm{d}}y} }}} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{f^{ - 1}}\left\{ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{1 - \mu }}} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{f^{ - 1}}\left\{ {f{{\left( a \right)}^{1 - \mu }}f{{\left( b \right)}^\mu }} \right\}. \end{array} $

下面讨论CQOWG算子的特殊情况.

(1) 当f(x)=kx, k≠0时, φ_{Q, f}([a, b])=b^μa^1-μ, 即CQOWG算子退化为COWG算子.

(2) 当f(x)=e^kx, k≠0时,

$ {\varphi _{Q,f}}\left( {\left[ {a,b} \right]} \right) = \mu b + \left( {1 - \mu } \right)a, $

即CQOWG算子退化为COWA算子.

(3) 当f(x)= ${{\rm{e}}^{\frac{x}{k}}}$, k≠0时,

φ_{Q, f}([a, b])=1/[(1-μ)/b+μ/b],

即CQOWG算子退化为COWH算子.

(4) 当Q(y)=y^r, r>0时,

$ {\varphi _{Q,f}}\left( {\left[ {a,b} \right]} \right) = {f^{ - 1}}\left[ {f{{\left( a \right)}^{\frac{r}{{r + 1}}}}f{{\left( b \right)}^{\frac{r}{{r + 1}}}}} \right]. $

此时, 若r=1, 则有

φ_{Q, f}([a, b])=f^-1[f(a)^1/2f(b)^1/2].

若r→0, 则有    φ_{Q, f}([a, b])=b.

若r→+∞, 则有    φ_{Q, f}([a, b])=a.

若r=k/K, 则有

$ {\varphi _{Q,f}}\left( {\left[ {a,b} \right]} \right) = {f^{ - 1}}\left[ {f{{\left( a \right)}^{\frac{k}{{k + K}}}}f{{\left( b \right)}^{\frac{k}{{k + K}}}}} \right]. $

下面讨论CQOWG算子的性质.

定理7(单调性)   设φ是CQOWG算子, 若a₁≤a₂, b₁≤b₂, 则对于任意BUM函数Q和导出函数f, 有φ_{Q, f}([a₁, b₁])≤φ_{Q, f}([a₂, b₂]).

证明   分2种情况证明.

(1) 当函数f(x)严格单调递增时, 函数f^-1(x)也严格单调递增, 则由a₁≤a₂, b₁≤b₂得f(a₁)≤f(a₂), f(b₁)≤f(b₂), 也即有f(a₁)^1-μ≤f(a₂)^1-μ, f(b₁)^μ≤f(b₂)^μ, 于是

$ f{\left( {{a_1}} \right)^{1 - \mu }}f{\left( {{b_1}} \right)^\mu } \le f{\left( {{a_2}} \right)^{1 - \mu }}f{\left( {{b_2}} \right)^\mu }, $

即有

$ {f^{ - 1}}\left[ {f{{\left( {{a_1}} \right)}^{1 - \mu }}f{{\left( {{b_1}} \right)}^\mu }} \right] \le {f^{ - 1}}\left[ {f{{\left( {{a_2}} \right)}^{1 - \mu }}f{{\left( {{b_2}} \right)}^\mu }} \right], $

所以有

$ {\varphi _{Q,f}}\left( {\left[ {{a_1},{b_1}} \right]} \right) \le {\varphi _{Q,f}}\left( {\left[ {{a_2},{b_2}} \right]} \right). $

(2) 当函数f(x)严格单调递减时, 同理可证

$ {\varphi _{Q,f}}\left( {\left[ {{a_1},{b_1}} \right]} \right) \le {\varphi _{Q,f}}\left( {\left[ {{a_2},{b_2}} \right]} \right). $

定理8(有界性)   设φ是CQOWG算子, 则a≤φ_{Q, f}([a, b])≤b.

证明  注意到当a=b时, φ_{Q, f}([a, b])=a, 然后由单调性可得

a=φ_{Q, f}([a, a])≤φ_{Q, f}([a, b])≤φ_{Q, f}([b, b])=b.

定理9(关于Q的单调性)   设φ是CQOWG算子, 若Q₁(x), Q₂(x)为BUM函数, 且x∈[0, 1], Q₁(x)≤Q₂(x), 则有

φ_{Q1, f}([a, b])≤φ_{Q2, f}([a, b]).

证明  令${\mu _1} = \int_0^1 {{Q_1}\left( y \right){\rm{d}}y,{\mu _2} = \int_0^1 {{Q_2}\left( y \right){\rm{d}}y} } $, 则由Q₁(x)≤Q₂(x)可知, 0≤μ₁≤μ₂≤1.

下面分2种情况证明.

(1) 当函数f(x)严格单调递增时, 函数f^-1(x)也严格单调递增, 则由a≤b得f(a)≤f(b), 进而有$\frac{{f\left( a \right)}}{{f\left( b \right)}} \ge 1$, 即

$ {\left( {\frac{{f\left( b \right)}}{{f\left( a \right)}}} \right)^{1 - {\mu _1}}} \ge {\left( {\frac{{f\left( b \right)}}{{f\left( a \right)}}} \right)^{1 - {\mu _2}}}, $

$ f\left( b \right){\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)^{1 - {\mu _1}}} \le f\left( b \right){\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)^{1 - {\mu _2}}}, $

即有

$ {f^{ - 1}}\left[ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{1 - {\mu _1}}}} \right] \le {f^{ - 1}}\left[ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{1 - {\mu _2}}}} \right], $

于是   φ_{Q1, f}([a, b])≤φ_{Q2, f}([a, b]).

(2) 当函数f(x)严格单调递减时, 同理可证φ_{Q1, f}([a, b])≤φ_{Q2, f}([a, b]).

定理10   设f(x)是任意的严格单调连续函数, 若对任意k>0, 有g(x)=f(x^k), 则有

$ {\varphi _{Q,g}}\left( {\left[ {a,b} \right]} \right) = {\left( {{\varphi _{Q,f}}\left( {\left[ {{a^k},{b^k}} \right]} \right)} \right)^{1/k}}. $

证明   令y=g(x)=f(x^k), 则有

x=g^-1(y)=(f^-1(y))^1/k,

于是有

$ \begin{array}{l} {\varphi _{Q,g}}\left( {\left[ {a,b} \right]} \right) = {g^{ - 1}}\left[ {g{{\left( a \right)}^{1 - \mu }}g{{\left( b \right)}^\mu }} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{g^{ - 1}}\left[ {{{\left( {f\left( {{a^k}} \right)} \right)}^{1 - \mu }}{{\left( {f\left( {{b^k}} \right)} \right)}^\mu }} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left( {{f^{ - 1}}\left[ {{{\left( {f\left( {{a^k}} \right)} \right)}^{1 - \mu }}\left( {f\left( {{b^k}} \right)} \right)\mu } \right]} \right)^{1/k}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left( {{\varphi _{Q,f}}\left( {\left[ {{a^k},{b^k}} \right]} \right)} \right)^{1/k}}. \end{array} $

定理11  设f(x)是任意的严格单调连续函数, g(x)是严格单调连续递增函数, 令h(x)=f(g(x)), 则φ_{Q, h}([a, b])=g^-1(φ_{Q, f}([g(a), g(b)])).

证明   令y=h(x)=f(g(x)), 则有x=h^-1(y)=g^-1(f^-1(y)), 于是

$ \begin{array}{l} {\varphi _{Q,h}}\left( {\left[ {a,b} \right]} \right) = {h^{ - 1}}\left[ {h{{\left( a \right)}^{1 - \mu }}h{{\left( b \right)}^\mu }} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{h^{ - 1}}\left[ {{{\left( {f\left( {g\left( a \right)} \right)} \right)}^{1 - \mu }}{{\left( {f\left( {g\left( b \right)} \right)} \right)}^\mu }} \right] = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{g^{ - 1}}\left\{ {{f^{ - 1}}\left[ {{{\left( {f\left( {g\left( a \right)} \right)} \right)}^{1 - \mu }}{{\left( {f\left( {g\left( b \right)} \right)} \right)}^\mu }} \right]} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{g^{ - 1}}\left( {{\varphi _{Q,f}}\left( {\left[ {g\left( a \right),g\left( b \right)} \right]} \right)} \right). \end{array} $

3 CQOWG算子的orness测度

受文献[25-27]启发, 本文将给出CQOWG算子的orness测度.

定义8   CQOWG算子的orness测度定义为

$ \begin{array}{*{20}{c}} {{\rm{ornes}}{{\rm{s}}_{Q,f}}\left( {\left[ {a,b} \right]} \right) = \frac{{{f^{ - 1}}\left\{ {f\left( b \right){{\left( {\frac{{f\left( a \right)}}{{f\left( b \right)}}} \right)}^{\int_0^1 {\frac{{{\rm{d}}Q\left( y \right)}}{{{\rm{d}}y}}y{\rm{d}}y} }}} \right\} - a}}{{b - a}} = }\\ {\frac{{{f^{ - 1}}\left[ {f{{\left( a \right)}^{1 - \mu }}f{{\left( b \right)}^\mu }} \right] - a}}{{b - a}}.} \end{array} $

显然, 当μ=0时, orness_{Q, f}([a, b])=0;当μ=1时, orness_{Q, f}([a, b])=1.

定理12   设φ_{Q, f}为任意的CQOWG算子, 则φ_{Q, f}([a, b])=a+(b-a)orness_{Q, f}([a, b]).

定理13   0≤orness_{Q, f}([a, b])≤1.

定理14   设orness_{Q, f}([a, b])是CQOWG算子_{Q, f}([(a, b)])的orness测度, 若Q₁(x), Q₂(x)为BUM函数, 且x∈[0, 1], 有Q₁(x)≤Q₂(x), 则有orness_{Q1, f}([a, b] )≤orness_{Q2, f}([a, b]).

定理15  设f(x)是任意的严格单调连续函数, Q(x)为BUM函数, 若g(x)=k(f(x))^c, k≠0, c≠0, 则φ_{Q, g}([a, b])=_{Q, f}([a, b]), orness_{Q, g}([a, b])=orness_{Q, f}([a, b]).

证明   令y=g(x)=k(f(x))^c, 则有

$x = {g^{ - 1}}y = {f^{ - 1}}\left( {{{\left( {\frac{y}{k}} \right)}^{1/c}}} \right)$, 于是,

$ \begin{array}{l} {\phi _{Q,g}}\left( {\left[ {a,b} \right]} \right) = {g^{ - 1}}\left[ {g{{\left( a \right)}^{1 - \mu }}g{{\left( b \right)}^\mu }} \right] = \\ \;\;\;\;\;\;\;\;{g^{ - 1}}\left[ {{{\left( {k{{\left( {f\left( a \right)} \right)}^c}} \right)}^{1 - \mu }}{{\left( {k{{\left( {f\left( b \right)} \right)}^c}} \right)}^\mu }} \right] = \\ \;\;\;\;\;\;\;\;{g^{ - 1}}\left[ {k{{\left( {f\left( a \right)} \right)}^{c\left( {1 - \mu } \right)}}{{\left( {f\left( b \right)} \right)}^{c\mu }}} \right] = \\ \;\;\;\;\;\;\;\;{f^{ - 1}}\left[ {{{\left( {\frac{{k{{\left( {f\left( a \right)} \right)}^{c\left( {1 - \mu } \right)}}{{\left( {f\left( b \right)} \right)}^{c\mu }}}}{k}} \right)}^{1/c}}} \right] = \\ \;\;\;\;\;\;\;\;{f^{ - 1}}\left[ {{{\left( {f\left( a \right)} \right)}^{1/\mu }}{{\left( {f\left( b \right)} \right)}^\mu }} \right] = {\phi _{Q,f}}\left( {\left[ {a,b} \right]} \right). \end{array} $

由CQOWG算子的orness测度定义可知,

orness_{Q, g}([a, b])=orness_{Q, f}([a, b]).

4 CQOWG算子的推广

CQOWG算子可以对单个连续区间数进行集成, 但不能集成多个连续区间数, 为此对其进行推广, 使之可以集合成2个或2个以上的连续区间数.

定义9   设[a_i, b_i](i=1, 2, …, n)为一组连续区间数, w=(w₁, w₂, …, w_n)为加权向量, 且w_i≥0, i=1, 2, …, n, $\sum\limits_{j = 1}^n {{w_j}} = 1$, 若函数γ:Ωⁿ→R⁺, $ {\gamma _w}\left( {\left[ {{a_1},{b_1}} \right],\left[ {{a_{2,}}{b_2}} \right],...,\left[ {{a_n},{b_n}} \right]} \right) = {\prod\limits_{i = 1}^n {\left( {{\varphi _{Q,f}}\left( {\left[ {{a_i},{b_i}} \right]} \right)} \right)} ^{{w_i}}}$则称γ为加权连续QOWG算子, 简称WCQOWG算子, 其中φ为CQOWG算子.

不难证明, WCQOWG算子具有下列性质:

定理16  设[a_i, b_i](i=1, 2, …, n)为一组连续区间数, w =(w₁, w₂, …, w_n)为加权向量, 且w_i≥0, i=1, 2, …, n, $\sum\limits_{j = 1}^n {{w_i}} = 1$, 则

(1) 幂等性

若a_i=a, b_i=b(i=1, 2, …, n), 则

γ_w([a₁, b₁], [a₂, b₂], …, [a_n, b_n])=φ_{Q, f}([a, b]).

(2) 有界性

$ \mathop {\min }\limits_i \left\{ {{a_i}} \right\} \le {\gamma _w}\left( {\left[ {{a_1},{b_1}} \right],\left[ {{a_2},{b_2}} \right], \cdots ,\left[ {{a_n},{b_n}} \right]} \right) \le \mathop {\max }\limits_i \left\{ {{b_i}} \right\}. $

(3) 单调性

设[c_i, d_i](i=1, 2, …, n)为另一组连续区间数, 若a_i≤c_i, b_i≤d_i(i=1, 2, …, n), 则

γ_w([a₁, b₁], [a₂, b₂], …, [a_n, b_n])≤γ_w([c₁, d₁], [c₂, d₂], …, [c_n, d_n]).

定义10   设[a_i, b_i](i=1, 2, …, n)为一组连续区间数, w=(w₁, w₂, …, w_n)为加权向量, 且w_j≥0, j=1, 2, …, n, $\sum\limits_{j = 1}^n {{w_j}} = 1$, 若函数

$ \eta :{\Omega ^n} \to {R^ + }, $

$ \begin{array}{*{20}{c}} {{\eta _\mathit{\boldsymbol{w}}}\left( {\left[ {{a_1},{b_1}} \right],\left[ {{a_2},{b_2}} \right], \cdots ,\left[ {{a_n},{b_n}} \right]} \right) = }\\ {\prod\limits_{j = 1}^n {{{\left( {{\varphi _{Q,f}}\left( {\left[ {{a_{\sigma \left( j \right)}},{b_{\sigma \left( j \right)}}} \right]} \right)} \right)}^{{w_j}}}} ,} \end{array} $

则称γ为有序加权连续QOWG算子, 简称为OWCQOWG算子, 其中φ为CQOWG算子, φ_{Q, f}([a_σ(j), b_σ(j)])为φ_{Q, f} ([a_i, b_i])(i=1, 2, …, n)中第j大的数.

不难证明, OWCQOWG算子具有下列性质:

定理17  设[a_i, b_i](i=1, 2, …, n)为一组连续区间数, w =(w₁, w₂, …, w_n)为加权向量, 且w_j≥0, j=1, 2, …, n, $\sum\limits_{j = 1}^n {{w_j}} = 1$, 则

(1) 幂等性

若a_i=a, b_i=b(i=1, 2, …, n), 则

η_w([a₁, b₁]), [a₂, b₂], …, [a_n, b_n] =φ_{Q, f}([a, b]).

(2) 有界性

$ \mathop {\min }\limits_i \left\{ {{a_i}} \right\} \le {\eta _w}\left( {\left[ {{a_1},{b_1}} \right],\left[ {{a_2},{b_2}} \right], \cdots ,\left[ {{a_n},{b_n}} \right]} \right) \le \mathop {\max }\limits_i \left\{ {{b_i}} \right\}. $

(3) 单调性

设[c_i, d_i](i=1, 2, …, n)为另一组连续区间数, 若a_i≤c_i, b_i≤d_i(i=1, 2, …, n), 则

η_w([a₁, b₁], [a₂, b₂], …, [a_n, b_n]）≤η_w([c₁, d₁], [c₂, d₂], …, [c_n, d_n]).

(4) 置换不变性

设[c_i, d_i](i=1, 2, …, n)为[a_i, b_i](i=1, 2, …, n)的任意一个置换, 则

η_w [a₁, b₁], [a₂, b₂], …, [a_n, b_n] =

η_w [c₁, d₁], [c₂, d₂], …, [c_n, d_n].

定义11   设[a_i, b_i](i=1, 2, …, n)为一组连续区间数, 若函数

$ \rho :{\Omega ^n} \to {R^ + }, $

$ \begin{array}{*{20}{c}} {{\rho _\mathit{\boldsymbol{w}}}\left( {\left[ {{a_1},{b_1}} \right],\left[ {{a_2},{b_2}} \right], \cdots ,\left[ {{a_n},{b_n}} \right]} \right) = }\\ {\prod\limits_{j = 1}^n {{{\left( {{\pi _{Q,f}}\left( {\left[ {{a_{\sigma \left( j \right)}},{b_{\sigma \left( j \right)}}} \right]} \right)} \right)}^{{w_j}}}} ,} \end{array} $

则称ρ为组合连续QOWG算子, 简称CCQOWG算子, 其中, (σ(1), σ(2), …, σ(n))是(1, 2, …, n)的一个置换, 满足

$ \begin{array}{l} {\pi _{Q,f}}\left( {\left[ {{a_{\sigma \left( 1 \right)}},{b_{\sigma \left( 1 \right)}}} \right]} \right) \ge \\ \;\;\;\;\;\;\;{\pi _{Q,f}}\left( {\left[ {{a_{\sigma \left( 2 \right)}},{b_{\sigma \left( 2 \right)}}} \right]} \right) \ge \cdots \ge \\ \;\;\;\;\;\;\;{\pi _{Q,f}}\left( {\left[ {{a_{\sigma \left( n \right)}},{b_{\sigma \left( n \right)}}} \right]} \right), \end{array} $

即π_{Q, f}([a_σ(j), b_σ(j)])为集合{(φ_{Q, f}([a₁, b₁]))nω₁),(φ_{Q, f}([a₂, b₂]))nω₂, …,(φ_{Q, f}([a_n, b_n]))nω_n}中第j大的数, φ为CQOWG算子, w=(w₁, w₂, …, w_n)为CCQOWG算子相关联的加权向量, 且w_j≥0, j=1, 2, …, n, $ \sum\limits_{j = 1}^n {{w_j}} = 1$, 而ω=(ω₁, ω₂, …, ω_n)为数据组[a_i, b_i](i=1, 2, …, n)的加权向量, 且ω_j≥0, j=1, 2, …, n, $\sum\limits_{j = 1}^n {{w_j}} = 1$, n为平衡因子.

显然, 当$\mathit{\boldsymbol{w = }}\left( {\frac{1}{n},\frac{1}{n},...,\frac{1}{n},} \right)$时, CCQOWG算子退化为WCQOWG算子; 当$\mathit{\boldsymbol{w = }}\left( {\frac{1}{n},\frac{1}{n},...,\frac{1}{n},} \right)$时, CCQOWG算子退化为OWCQOWG算子.

5 决策应用

在使用COWG算子进行区间数信息集成的过程中, 决策者既可以通过选择不同的生成函数实现对决策结果的调整, 也可通过选择态度参数体现决策的主动性, 因此COWG算子在区间数信息集成中具有独特的理论和应用优势.下面提出基于CQOWG算子的区间数多属性群决策方法.

设X ={x₁, x₂, …, x_m}为备选方案集, 属性集为U={u₁, u₂, …, u_n}, ω=(ω₁, ω₂, …, ω_n)为属性权重向量, 且ω_j≥0, j=1, 2, …, n, $\sum\limits_{j = 1}^n {{w_j}} = 1$.专家集合为D ={d₁, d₂, …, d_t}, μ=(μ₁, μ₂, …, μ_t)为专家权重向量, 且μ_j≥0, j=1, 2, …, t, $\sum\limits_{j = 1}^t {{\mu _j}} = 1$.专家d_k对方案x_i按属性u_j进行测度, 得到属性值为连续区间数a_ij^(k)=[a_ij^(k)-, a_ij^(k)+], 于是得到专家d_k的连续区间数决策矩阵M^(k)=(a_ij^(k))_mn.由于专家可能来自不同的领域, 其知识结构、实践经验、表达能力以及个人偏好等均不相同, 因此给出的决策矩阵也不相同, 为此通过选取适当的生成函数和态度参数, 使用CQOWG算子可将各专家决策矩阵集结成群体决策矩阵Z=(z_ij)_mn.然后, 由群体决策矩阵求出方案的综合属性值, 从而实现方案的择优排序.具体决策步骤如下:

步骤1   设决策者d_k给出的方案x_i在属性u_j下的测度为连续区间数a_ij^(k)=[a_ij^(k)-, a_ij^(k)+], 于是得到连续区间数决策矩阵M^(k)=(a_ij^(k))_mn, k=1, 2, …, t, i=1, 2, …, m, j=1, 2, …, n.

步骤2   由于属性可以分为效益型I₁和成本型I₂两类, 需要将决策矩阵M^(k)=(a_ij^(k))_mn规范为R^(k)=(r_ij^(k))_mn, r_ij^(k)=[r_ij^(k)-, r_ij^(k)+], 其中^[28],

$ \begin{array}{l} r_{ij}^{\left( k \right) - } = \frac{{a_{ij}^{\left( k \right) - }}}{{\sqrt {\sum\limits_{i = 1}^m {{{\left( {a_{ij}^{\left( k \right) + }} \right)}^2}} } }},r_{ij}^{\left( k \right) + } = \frac{{a_{ij}^{\left( k \right) + }}}{{\sqrt {\sum\limits_{i = 1}^m {{{\left( {a_{ij}^{\left( k \right) - }} \right)}^2}} } }},\\ \;\;\;\;\;\;j \in {I_1},i = 1,2, \cdots ,m; \end{array} $

$ \begin{array}{l} r_{ij}^{\left( k \right) - } = \frac{{1/a_{ij}^{\left( k \right) + }}}{{\sqrt {\sum\limits_{i = 1}^m {{{\left( {1/a_{ij}^{\left( k \right) - }} \right)}^2}} } }},r_{ij}^{\left( k \right) + } = \frac{{1/a_{ij}^{\left( k \right) - }}}{{\sqrt {\sum\limits_{i = 1}^m {{{\left( {a_{ij}^{\left( k \right) + }} \right)}^2}} } }},\\ \;\;\;\;\;\;j \in {I_2},i = 1,2, \cdots ,m. \end{array} $

步骤3   利用CCQOWG算子, 将t个连续区间数决策矩阵集成为综合决策矩阵Z=(z_ij)_mn, 其中,

$ \begin{array}{l} {z_{ij}} = \\ {\rho _\mathit{\boldsymbol{w}}}\left( {\left[ {r_{ij}^{\left( 1 \right) - },r_{ij}^{\left( 1 \right) + }} \right],\left[ {r_{ij}^{\left( 2 \right) - },r_{ij}^{\left( 2 \right) + }} \right], \cdots ,\left[ {r_{ij}^{\left( t \right) - },r_{ij}^{\left( t \right) + }} \right]} \right) = \\ \prod\limits_{k = 1}^t {{{\left( {{\pi _{Q,f}}\left( {\left[ {r_{ij}^{\sigma \left( k \right) - },r_{ij}^{\sigma \left( k \right) + }} \right]} \right)} \right)}^{{w_k}}}} ,\\ i = 1,2, \cdots ,m,j = 1,2, \cdots ,n. \end{array} $

步骤4  对综合决策矩阵Z=(z_ij)_mn进行集成, 得到每个方案的综合属性值${z_i}\left( \mathit{\boldsymbol{\omega }} \right) = \prod\limits_{j = 1}^n {z_{i{j^j}}^\omega } $,i=1,2,…,m.

步骤5  根据方案的综合属性值z_i(ω)(i=1, 2, …, m)实现方案{x₁, x₂, …, x_m}的择优排序.

例1^[25]  某投资银行要选择一家企业进行投资, 现有4家企业{x₁, x₂, x₃, x₄}可供选择, 决策者通过4个属性指标来评价投资价值, 即u₁:投资收益和负税率; u₂:投资净输出率; u₃:国际收益率; u₄:环境污染程度, 其中属性u₄为成本型指标, 其他属性为效益型指标, 属性权重向量为(0.30, 0.35, 0.15, 0.20).经过3位专家{d₁, d₂, d₃}的评估, 建立了区间数决策矩阵M^(k)=(a_ij^(k))_4×4, k=1, 2, 3, 专家权重向量为(0.4, 0.3, 0.3).下面根据专家提供的决策矩阵, 评价这4家企业的投资价值, 为该银行决策提供参考.

步骤1  建立区间数决策矩阵, 见表 1~表 3.

步骤2  将区间数决策矩阵M^(k)规范化, 得到规范化决策矩阵R^(k)(k=1, 2, 3).

$ \begin{array}{l} {\mathit{\boldsymbol{R}}^{\left( 1 \right)}} = \left( {\begin{array}{*{20}{c}} {\left[ {0.2537,0.5304} \right]}&{\left[ {0.3936,0.5704} \right]}\\ {\left[ {0.5286,0.8375} \right]}&{\left[ {0.4972,0.6944} \right]}\\ {\left[ {0.3594,0.6700} \right]}&{\left[ {0.3522,0.5208} \right]}\\ {\left[ {0.3171,0.5584} \right]}&{\left[ {0.4143,0.5952} \right]} \end{array}} \right.\\ \;\;\;\;\;\;\;\;\;\left. {\begin{array}{*{20}{c}} {\left[ {0.4135,0.5904} \right]}&{\left[ {0.4488,0.5089} \right]}\\ {\left[ {0.3675,0.5904} \right]}&{\left[ {0.3876,0.4214} \right]}\\ {\left[ {0.4135,0.6467} \right]}&{\left[ {0.5443,0.5994} \right]}\\ {\left[ {0.4365,0.6186} \right]}&{\left[ {0.5016,0.5619} \right]} \end{array}} \right), \end{array} $

$ \begin{array}{l} {\mathit{\boldsymbol{R}}^{\left( 2 \right)}} = \left( {\begin{array}{*{20}{c}} {\left[ {0.2229,0.4761} \right]}&{\left[ {0.4180,0.6262} \right]}\\ {\left[ {0.4661,0.8122} \right]}&{\left[ {0.4180,0.7623} \right]}\\ {\left[ {0.4053,0.6721} \right]}&{\left[ {0.3762,0.5445} \right]}\\ {\left[ {0.3040,0.7562} \right]}&{\left[ {0.3135,0.6534} \right]} \end{array}} \right.\\ \;\;\;\;\;\;\;\;\;\left. {\begin{array}{*{20}{c}} {\left[ {0.3371,0.6424} \right]}&{\left[ {0.4343,0.5522} \right]}\\ {\left[ {0.3852,0.5931} \right]}&{\left[ {0.3536,0.4314} \right]}\\ {\left[ {0.4334,0.5931} \right]}&{\left[ {0.4951,0.6135} \right]}\\ {\left[ {0.4575,0.6228} \right]}&{\left[ {0.4951,0.6135} \right]} \end{array}} \right), \end{array} $

$ \begin{array}{l} {\mathit{\boldsymbol{R}}^{\left( 3 \right)}} = \left( {\begin{array}{*{20}{c}} {\left[ {0.2810,0.7653} \right]}&{\left[ {0.3812,0.5762} \right]}\\ {\left[ {0.4683,0.4917} \right]}&{\left[ {0.4036,0.5238} \right]}\\ {\left[ {0.2810,0.6597} \right]}&{\left[ {0.4260,0.6024} \right]}\\ {\left[ {0.3559,0.5806} \right]}&{\left[ {0.4933,0.6285} \right]} \end{array}} \right.\\ \;\;\;\;\;\;\;\;\;\left. {\begin{array}{*{20}{c}} {\left[ {0.4531,0.5897} \right]}&{\left[ {0.4961,0.5614} \right]}\\ {\left[ {0.2719,0.5897} \right]}&{\left[ {0.3898,0.4649} \right]}\\ {\left[ {0.4078,0.5897} \right]}&{\left[ {0.4787,0.5510} \right]}\\ {\left[ {0.4531,0.7021} \right]}&{\left[ {0.4625,0.5950} \right]} \end{array}} \right). \end{array} $

步骤3  利用CCQOWG算子, 求出综合决策矩阵:

$ \mathit{\boldsymbol{Z}} = \left( {\begin{array}{*{20}{c}} {0.4441}&{0.5588}&{0.5555}&{0.5432}\\ {0.6973}&{0.6081}&{0.5255}&{0.4434}\\ {0.5808}&{0.5217}&{0.5603}&{0.5863}\\ {0.5617}&{0.5575}&{0.5984}&{0.5886} \end{array}} \right). $

其中, $Q\left( x \right) = \sqrt x $, f(x)=x², CCQOWG算子的相关联加权向量为w=(0.067, 0.666, 0.267), 专家权重向量为(0.4, 0.3, 0.3).

步骤4  方案综合属性值分别为

z₁(ω)=0.518 2, z₂(ω)=0.581 9,

z₃(ω)=0.557 4, z₄(ω)=0.570 9,

其中, 属性权重向量ω=(0.30, 0.35, 0.15, 0.20).

步骤5  根据方案综合属性值得到方案的排序为x₂＞x₄＞x₃＞x₁, 即方案x₂最优.

使用文献[25]中的CCQOWA算子, 计算得各方案综合属性值为z₁(ω)=0.572 7, z₂(ω)=0.654 9, z₃(ω)=0.607 6, z₄(ω)=0.622 0, 因此方案排序为x₂＞x₄＞x₃＞x₁.

尽管各方案的综合属性值与使用本文中的CCQOWG算子得到的综合属性值不同, 但是方案的排序完全相同.这在一定程度上说明了使用本文的CCQOWG算子进行决策是有效的.

6 结语

首先, 通过拓展QOWG算子, 提出了连续区间QOWG算子, 拓展了QOWG算子的应用范围, 研究了CQOWG算子的特殊情况和性质.其次, 定义了CQOWG算子的orness测度, 研究了orness测度的性质.然后, 定义了WCQOWG算子、OWCQOWG算子以及CCQOWG算子, 使得CQOWG算子可以处理多个连续区间数的集成问题, 讨论了这些算子的性质.最后, 提出了基于连续QOWG算子的多属性群决策方法, 并通过决策实例说明了其可行性和有效性.