Answers.Body: <p>The square brackets <span class="math-container">$[]$</span> in <span class="math-container">$[\tau_{ij}]^\alpha$</span> and <span class="math-container">$[\eta_{ij}]^\beta$</span> may be just a way of emphasing that the elements <span class="math-container">$\tau_{ij} \in \mathbb{R}$</span> and <span class="math-container">$\eta_{ij} \in \mathbb{R}$</span> of respectively the matrices <span class="math-container">$\mathbf{\tau} \in \mathbb{R}^{n \times n}$</span> and <span class="math-container">$\mathbf{\eta} \in \mathbb{R}^{n \times n}$</span> (where <span class="math-container">$n$</span> is the number of nodes in the graph) are respectively raised to <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>, so they could have used also other type of brackets, for example, <span class="math-container">$()$</span>. It may also be a way of indicating that <span class="math-container">$[\tau_{ij}]$</span> and <span class="math-container">$[\eta_{ij}]$</span> are <span class="math-container">$1 \times 1$</span> matrices or vectors that contain respectively the scalars <span class="math-container">$\tau_{ij}$</span> and <span class="math-container">$\eta_{ij}$</span>, so you are multiplying matrices or vectors (dot product). </p>
  
  <p>This notation is also used in the paper that introduced the <em>ant colony system</em> (ACS) (and it is probably used in many other papers related to ant colony optimization). See equation 1 of <a href="http://people.idsia.ch/~luca/acs-ec97.pdf" rel="nofollow noreferrer">Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem</a> (1997) by Dorigo and Gambardella.</p>
  
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Answers.CreationDate: 2019-11-01T16:38:47.700Z
Answers.Text: By the way, in the past, I've implemented the ant colony system (ACS). I've looked at my implementation and it seems to be consistent with my explanations above.
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Answers.CreationDate: 2019-11-01T14:56:44.023Z
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Body: <p>I'm studying the paper &quot;<a href="https://pdfs.semanticscholar.org/f1b8/afdbf9f108afdb42476850ed3fc13db43cea.pdf" rel="nofollow noreferrer">Minimizing Total Tardiness on a Single Machine Using Ant Colony Optimization</a>&quot; which has proposed to use Ant colony optimization to SMTWTP.</p>
  <p>According to this paper:</p>
  <blockquote>
  <p>Each artificial ant iteratively and independently decides which job to
  append to the sub-sequence generated so far until all jobs are
  scheduled, Each ant generates a complete solution by selecting a job <span class="math-container">$j$</span>
  to be on the <span class="math-container">$i$</span>-th position of the sequence. This selection process is
  influenced through problem-specific heuristic information called
  visibility and denoted by <span class="math-container">$\eta_{ij}$</span> as well as pheromone trails denoted by <span class="math-container">$\tau_{ij}$</span>. The former is an indicator of how good the choice of that job
  seems to be and the latter indicates how good the choice of that job
  was in former runs. Both matrices are only two dimensional as a
  consequence of the reduction in complexity</p>
  </blockquote>
  <p>They have proposed this formula for the probability that job <span class="math-container">$j$</span> be selected to be processed on position <span class="math-container">$i$</span> (page 9 of the linked paper):</p>
  <p><span class="math-container">$$
  \mathcal{P}_{i j}=\left\{\begin{array}{cl}
  \frac{\left[\tau_{i j}\right]^{\alpha}\left[\eta_{i j}\right]^{\beta}}{\sum_{h \in \Omega}\left[\tau_{i h}\right]^{\alpha}\left[\eta_{i h}\right]^{\beta}} &amp; \text { if } j \in \Omega \\
  0 &amp; \text { otherwise }
  \end{array}\right.\tag{1}\label{1}
  $$</span></p>
  <p>but I can't understand what <span class="math-container">$[]$</span> surrounding <span class="math-container">$\eta_{ij}$</span> and <span class="math-container">$\tau_{ij}$</span> indicates. Does it show that these values are matrices?</p>
  
Tags: <papers><swarm-intelligence><notation><ant-colony-optimization>
Title: What is the meaning of the square brackets in ant colony optimization?
UserDisplayName: Pablo
ThreadText: # What is the meaning of the square brackets in ant colony optimization?
  
  (Asked by: username_0 on 2019-11-01)
  
  I'm studying the paper "[Minimizing Total Tardiness on a Single Machine Using Ant Colony Optimization](https://pdfs.semanticscholar.org/f1b8/afdbf9f108afdb42476850ed3fc13db43cea.pdf)" which has proposed to use Ant colony optimization to SMTWTP.
  
  According to this paper:
  
  > Each artificial ant iteratively and independently decides which job to
  > append to the sub-sequence generated so far until all jobs are
  > scheduled, Each ant generates a complete solution by selecting a job $j$
  > to be on the $i$-th position of the sequence. This selection process is
  > influenced through problem-specific heuristic information called
  > visibility and denoted by $\eta\_{ij}$ as well as pheromone trails denoted by $\tau\_{ij}$. The former is an indicator of how good the choice of that job
  > seems to be and the latter indicates how good the choice of that job
  > was in former runs. Both matrices are only two dimensional as a
  > consequence of the reduction in complexity
  
  They have proposed this formula for the probability that job $j$ be selected to be processed on position $i$ (page 9 of the linked paper):
  
  $$
  \mathcal{P}\_{i j}=\left\{\begin{array}{cl}
  \frac{\left[\tau\_{i j}\right]^{\alpha}\left[\eta\_{i j}\right]^{\beta}}{\sum\_{h \in \Omega}\left[\tau\_{i h}\right]^{\alpha}\left[\eta\_{i h}\right]^{\beta}} & \text { if } j \in \Omega \\
  0 & \text { otherwise }
  \end{array}\right.\tag{1}\label{1}
  $$
  
  but I can't understand what $[]$ surrounding $\eta\_{ij}$ and $\tau\_{ij}$ indicates. Does it show that these values are matrices?
  
  **Tags:** papers, swarm-intelligence, notation, ant-colony-optimization
  
  
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  ## 1 answers
  
  ### Answer by username_1 on 2019-11-01. Score: 3 | Accepted answer
  
  The square brackets $[]$ in $[\tau\_{ij}]^\alpha$ and $[\eta\_{ij}]^\beta$ may be just a way of emphasing that the elements $\tau\_{ij} \in \mathbb{R}$ and $\eta\_{ij} \in \mathbb{R}$ of respectively the matrices $\mathbf{\tau} \in \mathbb{R}^{n \times n}$ and $\mathbf{\eta} \in \mathbb{R}^{n \times n}$ (where $n$ is the number of nodes in the graph) are respectively raised to $\alpha$ and $\beta$, so they could have used also other type of brackets, for example, $()$. It may also be a way of indicating that $[\tau\_{ij}]$ and $[\eta\_{ij}]$ are $1 \times 1$ matrices or vectors that contain respectively the scalars $\tau\_{ij}$ and $\eta\_{ij}$, so you are multiplying matrices or vectors (dot product).
  
  This notation is also used in the paper that introduced the *ant colony system* (ACS) (and it is probably used in many other papers related to ant colony optimization). See equation 1 of [Ant Colony System: A Cooperative Learning Approach to the Traveling Salesman Problem](http://people.idsia.ch/~luca/acs-ec97.pdf) (1997) by Dorigo and Gambardella.
  
  > **Comments:**
  > **username_1**: By the way, in the past, I've implemented the ant colony system (ACS). I've looked at my implementation and it seems to be consistent with my explanations above.  
  
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