基于后悔理论和TOPSIS的灰色随机MCDM方法

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　　本期为您带来的是＂基于后悔理论和TOPSIS的灰色随机MCDM方法＂
　　多属性决策.mp3  音频：  00:00  /  05:02
　　1
　　决策问题的描述
　　考虑以下灰色随机MCDM问题：假设Ai是m个可能备选方案的离散备选方案集，Cj是n个标准的集合。标准的加权向量为Wj，其中：
　　由于决策环境的不确定性，备选方案将具有某种可能的状态。
　　灰色随机决策矩阵可以表示为
　　然后，我们能够根据提供的信息对备选方案进行排序。
　　2
　　决策步骤
　　第1步：规范化决策矩阵
　　为了消除物理量不同维度的影响，需要对决策矩阵进行归一化，以便将各种标准值转换为可比值。这些准则通常分为两类：最大化准则和最小化准则。如果标准为最大化类型，则转换公式如下：
　　其中，
　　如果标准为最小化类型，则公式如下：
　　其中：
　　归一化决策矩阵表示为：
　　第2步：确定理想点。
　　第3步：计算与标准相关的效用值和后悔值。
　　（a）应在计算标准的效用值之前构建效用函数。由于决策者的风险厌恶，效用函数v(x)是单调递增的凹函数，其中：
　　这里，幂函数用作效用函数的标准值。
　　如果风险规避系数越小，决策者的风险厌恶越大。
　　这里，考虑了两种分布。
　　(1) 均匀分布。均匀分布是最常见的分布之一。对于服从均匀分布的灰色随机变量x，概率密度函数为
　　（2）正态分布。正态分布在统计学中非常重要，在自然科学和社会科学中经常用于具有未知分布的实值随机变量。对于服从正态分布的灰色随机变量x，概率密度函数为：
　　这里，平均值为
　　标准偏差为
　　（b）后悔函数用于确定与标准有关的后悔值。它是单调递增的凹函数，并且满足：
　　可表示如下：
　　当R（u（a）-u（b））>0时，表示决策者对选择方案A舍弃方案B感到欣喜；当R（u（a）-u（b））<0时，表示决策者对选择方案A舍弃方案B感到后悔。
　　与理想点相比，在第t中状态下，备选方案ai相对于cj的后悔值可计算如下：
　　第4步：计算备选方案的总体感知效用值。
　　可根据步骤3计算备选方案的感知效用值。这些代表了备选方案的效用价值和后悔价值之和。令感知效用Uij为第t状态下备选方案ai相对于cj的感知效用值，因此，
　　备选方案ai相对于cj的整体感知效用价值可计算如下：
　　其中，Pj是更精确的区间概率，可通过以下公式计算：
　　整体感知效用矩阵表示如下：
　　第5步：根据总体感知效用对备选方案进行优先级排序
　　确定整体感知效用区间的正理想解和负理想解，如下所示：
　　计算每个总体感知效用区间与正理想解和负理想解之间的距离：
　　然后，用以下公式估算相对接近度：
　　其中，Ci越大，方案ai越好。
　　英文学习：
　　Description of the decision problem
　　Consider the following gray random MCDM problem:
　　Suppose Ai is a discrete set of m possible alternatives, and Cj is a set of n criteria. The standard weighting vector is Wj, where:
　　Due to the uncertainty of the decision-making environment, the alternatives will have a certain possible state.
　　The gray random decision matrix can be expressed as
　　We can then sort the alternatives based on the information provided.
　　Decision steps
　　For the above gray random MCDM problem, the solution process can be summarized as follows.
　　Step 1: Standardize the decision matrix
　　In order to eliminate the influence of different dimensions of physical quantities, it is necessary to normalize the decision matrix to convert various standard values into comparable values. These criteria are usually pided into two categories: maximization criteria and minimization criteria. If the standard is maximized, the conversion formula is as follows:
　　in,
　　If the standard is a minimized type, the formula is as follows:
　　in:
　　The normalized decision matrix is expressed as:
　　Step 2: Determine the ideal point.
　　Step 3: Calculate the utility value and regret value related to the standard.
　　(A) The utility function should be constructed before calculating the utility value of the standard. Due to the risk aversion of decision makers, the utility function v(x) is a monotonically increasing concave function, where
　　Here, the power function is used as the standard value of the utility function.
　　If the risk aversion coefficient is smaller, the risk aversion of the decision maker is greater.
　　Here, two distributions are considered.
　　(1) Evenly distributed. Uniform distribution is one of the most common distributions. For a uniformly distributed gray random variable x, the probability density function is
　　(2) Normal distribution. Normal distribution is very important in statistics, and is often used for real-valued random variables with unknown distributions in natural sciences and social sciences. For a gray random variable x that obeys a normal distribution, the probability density function is:
　　Here, the average is
　　The standard deviation is
　　(B) The regret function is used to determine the regret value related to the standard. It is a monotonically increasing concave function and satisfies:
　　It can be expressed as follows:
　　When R(u(a)-u(b))>0, it means that the decision-maker is happy to choose option A and abandon option B; when R(u(a)-u(b))<0, it means that the decision-maker is happy Regret for choosing option A and abandoning option B.
　　Compared with the ideal point, in the t-th state, the regret value of alternative ai relative to cj can be calculated as follows:
　　Step 4: Calculate the overall perceived utility value of the alternatives.
　　The perceived utility value of the alternative can be calculated according to step 3. These represent the sum of the utility value and regret value of the alternatives. Let the perceived utility Uij be the perceived utility value of the alternative ai relative to cj in the t-th state, therefore,
　　The overall perceived utility value of alternative ai relative to cj can be calculated as follows:
　　Among them, Pj is a more accurate interval probability, which can be calculated by the following formula:
　　The overall perceived utility matrix is expressed as follows
　　Step 5: Prioritize alternatives based on the overall perceived utility interval
　　Determine the positive ideal solution and negative ideal solution of the overall perceived utility interval, as shown below:
　　Calculate the distance between each overall perceived utility interval and the positive ideal solution and the negative ideal solution:
　　Then, use the following formula to estimate the relative proximity:
　　The larger the Ci, the better the scheme ai.
　　英文翻译：谷歌翻译
　　参考资料：
　　[1]Zhou H , Wang J Q , Zhang H Y . Grey stochastic multi-criteria decision making based on regret theory and TOPSIS[J]. International Journal of Machine Learning and Cybernetics, 2015, 8(2):1-14.
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