顺序优先法

顺序优先法（OPA）是一种多准则决策分析方法（英语：Multiple-criteria decision analysis）（multi-criteria decision-making ,MCDM），有助于解决具有偏好关系的集體決策问题。

描述

大多数的多准则决策分析方法，如层次分析法（analytic hierarchy Process, AHP）和网络分析法（英语：Analytic network process）（Analytic Network Process, ANP），是以成对比较矩阵为基础的^[1]。

决策问题^[2]

该方法使用線性規劃方法同时计算专家、评价指标和备选方案的权重^[2]。在OPA方法中使用序数数据（英语：Ordinal data）的主要原因是与涉及人类的群体决策问题中使用的精确比例相比，序数数据的可及性和准确性^[3]。

在现实世界中，专家们可能对某一选择或评价指标没有足够的了解。这种情况下，问题的输入数据是不完整的，此时需要在OPA线性规划模型中删除与评价指标或备选方案相关的约束条件^[4]。

近年来，各种类型的数据归一化方法被应用于多准则决策方法 (multi-criteria decision-making ,MCDM) 中。Palczewski和 Satabun表明，使用各种数据归一化方法可以改变多准则决策方法的最终排名^[5]。Javed 及其同事表明，可以通过避免数据归一化来解决多准则决策问题^[6]。不需要对偏好关系进行归一化，因此，OPA方法不需要数据归一化^[7]。

OPA方法

OPA模型是一个线性规划模型，可以利用单纯形法来解决。该方法的步骤如下:^[8][9][2]

第一步: 确定专家，并根据工作经验、教育资格等确定专家的优先次序。

第二步: 确定评价指标，并确定每个专家对指标的偏好。

第三步: 确定备选方案，并由每个专家确定在每一评价指标下备选方案的偏好。

第四步: 构建以下线性规划模型，并通过适当的优化软件如LINGO、GAMS、MATLAB等进行求解。

${\textstyle {\begin{aligned}&MaxZ\\&S.t.\\&Z\leq r_{i}{\bigg (}r_{j}{\big (}r_{k}(w_{ijk}^{r_{k}}-w_{ijk}^{{r_{k}}+1}){\big )}{\bigg )}\;\;\;\;\forall i,j\;and\;r_{k}\\&Z\leq r_{i}r_{j}r_{m}w_{ijk}^{r_{m}}\;\;\;\forall i,j\;and\;r_{m}\\&\sum _{i=1}^{p}\sum _{j=1}^{n}\sum _{k=1}^{m}w_{ijk}=1\\&w_{ijk}\geq 0\;\;\;\forall i,j\;and\;k\\&Z:Unrestricted\;in\;sign\\\end{aligned}}}$

在上述模型中。${\displaystyle r_{i}(i=1,...,p)}$代表专家的等级${\displaystyle i}$, ${\displaystyle r_{j}(j=1...,n)}$代表指标的等级${\displaystyle j}$,${\displaystyle r_{k}(k=1...,m)}$代表备选方案的等级${\displaystyle k}$。而${\displaystyle w_{ijk}}$代表专家i在评价指标j下备选方案k的权重。在解决OPA线性规划模型后，每个备选方案的权重由以下公式计算。

${\displaystyle {\begin{aligned}&w_{k}=\sum _{i=1}^{p}\sum _{j=1}^{n}w_{ijk}\;\;\;\;\forall k\\\end{aligned}}}$

每个评价指标的权重按以下公式计算。

${\displaystyle {\begin{aligned}&w_{j}=\sum _{i=1}^{p}\sum _{k=1}^{m}w_{ijk}\;\;\;\;\forall j\\\end{aligned}}}$

每个专家的权重按以下公式计算。

${\displaystyle {\begin{aligned}&w_{i}=\sum _{j=1}^{n}\sum _{k=1}^{m}w_{ijk}\;\;\;\;\forall i\\\end{aligned}}}$

例子

例子的决策问题

假设要调查买房子的问题^[10]。在这个决策问题中，有两位专家，同时有两个评价指标，即成本（c）和建筑质量（q），为房屋的选择提供标准。另一方面，有三所房子(h1，h2，h3）可供购买。第一个专家（x）有三年的工作经验，第二个专家（y）有两年的工作经验。该问题的结构如图所示。

第 1 步：第一位专家（x）比专家（y）有更多经验，因此 x>y。

第 2 步：专家对评价指标的偏好总结在下表中。

专家对评价指标的意见

评价指标	专家（x）	专家（y）
c	1	2
q	2	1

第 3 步：专家对备选方案的偏好总结在下表中。

专家对备选方案的意见

备选方案	专家（x)		专家（y）
	c	q	c	q
h1	1	2	1	3
h2	3	1	2	1
h3	2	3	3	2

第 4 步：根据输入数据形成 OPA 线性规划模型，具体如下。

${\displaystyle {\begin{aligned}&MaxZ\\&S.t.\\&Z\leq 1*1*1*(w_{xch1}-w_{xch3})\;\;\;\;\\&Z\leq 1*1*2*(w_{xch3}-w_{xch2})\;\;\;\;\\&Z\leq 1*1*3*w_{xch2}\;\;\;\\\\&Z\leq 1*2*1*(w_{xqh2}-w_{xqh1})\;\;\;\;\\&Z\leq 1*2*2*(w_{xqh1}-w_{xqh3})\;\;\;\;\\&Z\leq 1*2*3*w_{xqh3}\;\;\;\\\\&Z\leq 2*2*1*(w_{ych1}-w_{ych2})\;\;\;\;\\&Z\leq 2*2*2*(w_{ych2}-w_{ych3})\;\;\;\;\\&Z\leq 2*2*3*w_{ych3}\;\;\;\\\\&Z\leq 2*1*1*(w_{yqh2}-w_{yqh3})\;\;\;\;\\&Z\leq 2*1*2*(w_{yqh3}-w_{yqh1})\;\;\;\;\\&Z\leq 2*1*3*w_{yqh1}\;\;\;\\\\&w_{xch1}+w_{xch2}+w_{xch3}+w_{xqh1}+w_{xqh2}+w_{xqh3}+w_{ych1}+w_{ych2}+w_{ych3}+w_{yqh1}+w_{yqh2}+w_{yqh3}=1\\\\\end{aligned}}}$

用优化软件求解上述模型后，得到专家、评价指标和备选方案的权重如下。

${\displaystyle {\begin{aligned}&w_{x}=w_{xch1}+w_{xch2}+w_{xch3}+w_{xqh1}+w_{xqh2}+w_{xqh3}=0.666667\\\\&w_{y}=w_{ych1}+w_{ych2}+w_{ych3}+w_{yqh1}+w_{yqh2}+w_{yqh3}=0.333333\\\\\\&w_{c}=w_{xch1}+w_{xch2}+w_{xch3}+w_{ych1}+w_{ych2}+w_{ych3}=0.555556\\\\&w_{q}=w_{xqh1}+w_{xqh2}+w_{xqh3}+w_{yqh1}+w_{yqh2}+w_{yqh3}=0.444444\\\\\\&w_{h1}=w_{xch1}+w_{xqh1}+w_{ych1}+w_{yqh1}=0.425926\\\\&w_{h2}=w_{xch2}+w_{xqh2}+w_{ych2}+w_{yqh2}=0.351852\\\\&w_{h3}=w_{xch3}+w_{xqh3}+w_{ych3}+w_{yqh3}=0.222222\\\\\end{aligned}}}$

因此，房子1（h1）被认为是最佳选择。此外，可以认为，评价指标成本（c）比评价指标建筑质量（q）更重要。另外，根据专家的权重，可以认为，与专家（y）相比，专家（x）对最终选择的影响更大。

应用

OPA方法在各个研究领域的应用总结如下。

农业、制造业、服务业

• 制造业供应链^[11][6]

• 可持续农业^[12]
• 生产战略^[13]
• 生产调度^[14]
• 汽车工业^[15]
• 社区服务需求^[16]

建筑行业

• 建筑分包^[17]
• 可持续建造^{[7][18][19][20][21]}
• 项目管理^[9][22][4]

能源与环境

• 太阳能和风能^[23]
• 低碳技术^[24]
• 电气化和排放^[25]
• 循環經濟^[26]

医疗保健

• 2019冠状病毒病^[27][28]
• 医疗保健供应链^[29]
• 社区服务^[30]

信息技术

• 元宇宙^[31][32]
• 自动驾驶汽车^[33]
• 过程控制^[34]
• 区块链^[18][19][26]
• 技术需求^[35]

交通运输

• 供应链管理^{[8][21][28][7]}
• 交通管制^[36]
• 道路维护^[37]

延伸

以下是 OPA 方法的几个扩展。

• 灰色顺序优先法 (OPA-G)^[7]
• 模糊顺序优先法 (OPA-F)^[28]
• OPA 中的置信度测量^[8]
• 鲁棒顺序优先法 (OPA-R)^[9]
• 混合 OPA-模糊 EDAS^[13]
• 混合 DEA-OPA 模型^[11]
• 混合型 MULTIMOORA-OPA^[38]
• 团体加权顺序优先法 (GWOPA)^[39]

软件

以下非盈利工具可用于解决使用 OPA 方法的 MCDM 问题。

• 基于网络的解算器^[40]
• 基于 Excel 的解算器^[41]
• 基于林格的解算器^[42]
• 基于 Matlab 的求解器^[43]

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