用复杂指标写出多个和的更好方法

Question

有没有办法以更清晰或更易读的方式显示多个系列的索引？我很高兴收到任何建议。

\documentclass{report}
 \usepackage{amsmath}
\begin{document}
\begin{align}
\begin{split}
    & \sum_{\{j=1,\dots ,m\}}
      \sum_{\{p'\in X_{j}|s-l_{j}+1 \leq p' \leq s\}}
      \sum_{\{q'\in Y_{j}|t-w_{j}+1 \leq q' \leq t\}}
      \sum_{\{r'\in Z_{j}|u+1 \leq r' \leq H-h_{j}\}}
      \left( \frac{P_{j}}{l_{j}\cdot w_{j}} \right) 
      \cdot x_{jp'q'r'} \\[10pt]
    & \leq \sum_{\{i=1,\dots ,m\}}
      \sum_{\{p\in X_{i}|s-l_{i}+1 \leq p \leq s\}}
      \sum_{\{q\in Y_{i}|t-w_{i}+1 \leq q \leq t\}}
      \sum_{\{r\in Z_{i}|u-h_{i}+1 \leq r \leq u\}}
      \sigma_{i} \cdot x_{ipqr} \\[10pt]
    & \forall s\in X, t\in Y, u\in Z.
\end{split}
\end{align}
\end{document}

Answer 1

与@David Carlisle 的想法相同，但有一些改进mathtools：

\documentclass{report}
\usepackage{mathtools}

\begin{document}

\begin{multline}
\sum_{\substack{
j=1,\dots ,m\\
\mathclap{p'\in X_{j},\: s-l_{j}+1 \leq p' \leq s}\\
\mathclap{q'\in Y_{j},\: t-w_{j}+1 \leq q' \leq t}\\
\mathclap{r'\in Z_{j},\: u+1 \leq r' \leq H-h_{j}}
}}\mkern-6mu
      \left( \frac{P_{j}}{l_{j}\cdot w_{j}} \right)
      \cdot x_{jp'q'r'}
     \leq\sum_{\substack{
i=1,\dots ,m\\
\mathclap{p\in X_{i},\: s-l_{i}+1 \leq p \leq s}\\
\mathclap{q\in Y_{i},\: t-w_{i}+1 \leq q \leq t}\\
\mathclap{r\in Z_{i},\: u-h_{i}+1 \leq r \leq u}
}}\mkern-12mu
      \sigma_{i} \cdot x_{ipqr} \\%[10pt]
    \forall s\in X, t\in Y, u\in Z.
\end{multline}

\end{document} 

Answer 2

如果我理解的正确，你能不能不使用单一总和？

\documentclass{report}
\usepackage{amsmath}
\begin{document}
\begin{multline}
\sum_{\substack{
j=1,\dots ,m\\
p'\in X_{j}\mid s-l_{j}+1 \leq p' \leq s\\
q'\in Y_{j}\mid t-w_{j}+1 \leq q' \leq t\\
r'\in Z_{j}\mid u+1 \leq r' \leq H-h_{j}
}}
      \left( \frac{P_{j}}{l_{j}\cdot w_{j}} \right) 
      \cdot x_{jp'q'r'} 
     \leq\sum_{\substack{
i=1,\dots ,m\\
p\in X_{i}\mid s-l_{i}+1 \leq p \leq s\\
q\in Y_{i}\mid t-w_{i}+1 \leq q \leq t\\
r\in Z_{i}\mid u-h_{i}+1 \leq r \leq u
}}
      \sigma_{i} \cdot x_{ipqr} \\[10pt]
    \forall s\in X, t\in Y, u\in Z.
\end{multline}
\end{document}

Answer 3

那这个呢？

\documentclass{report}
\usepackage{amsmath}
\begin{document}
\begin{align*}
&\sum_{j=1}^m\
 \sum_{\substack{p'=s-l_j+1\\p'\in X_j}}^s\
 \sum_{\substack{q'=t-w_j+1\\q'\in Y_j}}^t\
 \sum_{\substack{r'=u+1\\r'\in Z_j}}^{H-h_j}
 \frac{P_jx_{jp'q'r'}}{l_jw_j}\\
&\leq\sum_{i=1}^m\
  \sum_{\substack{p=s-l_i+1\\p\in X_i}}^s\
  \sum_{\substack{q=t-w_i+1\\q\in Y_i}}^t\
  \sum_{\substack{r=u-h_i+1\\r\in Z_i}}^u
  \sigma_ix_{ipqr} 
  \quad\forall s\in X,\ t\in Y,\ u\in Z.
\end{align*}
\end{document}