Answers.Body: <p>Let's start by making clear that both AS and MMAS use only global pheromone update. Now, the MMAS has two main differences regarding AS:</p>
  <ol>
  <li><p>In AS, all ants that completed a solution are used for the update, while in MMAS only the best ant with a complete solution is used for update (as you had pointed out).</p>
  </li>
  <li><p>In AS, the pheromone values are not explicitly bounded. In MMAS, the pheromones are enforced to lie within a pre-set interval <span class="math-container">$\tau_{min} \leq \tau_{ij} \leq \tau_{max}$</span> (which gives its name to the algorithm). To ensure this condition, the pheromone update is done by means of the formula</p>
  </li>
  </ol>
  <p><span class="math-container">$$
  \tau_{ij} \leftarrow \left[ (1-\rho) \cdot \tau_{ij} + \Delta \tau_{ij}^{best} \right]_{\tau_{min}}^{\tau_{max}},
  $$</span></p>
  <p>with the operator <span class="math-container">$[x]_a^b$</span> defined as</p>
  <p><span class="math-container">$$
  [x]_a^b= 
  \begin{cases}
      a \qquad \mathrm{if} \quad x &gt; a\\
      b \qquad \mathrm{if} \quad x &lt; b\\
      x \qquad \mathrm{otherwise}
  \end{cases}
  $$</span></p>
  <p>Reference: <a href="https://courses.cs.ut.ee/all/MTAT.03.238/2011K/uploads/Main/04129846.pdf" rel="nofollow noreferrer">Ant Colony Optimization. Artificial Ants as a Computational Intelligence Technique</a>.</p>
  
Answers.ContentLicense: CC BY-SA 4.0
Answers.CreationDate: 2020-02-08T11:22:46.520Z
Answers.LastActivityDate: 2021-01-15T11:39:35.903Z
Answers.LastEditDate: 2021-01-15T11:39:35.903Z
Answers.UserDisplayName: djbetancourt
Body: <p>I'm studying ant colony optimization. I'm trying to understand the difference between the ant system (AS) and the max-min ant system (MMAS) approaches. As far as I found out, the main difference between these 2 is that in AS the pheromone trail is updated after all ants have finished the tour (it means all ants participate in this update), but in MMAS, only the best ant updates this value. Am I right? Is there any other significant difference?</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2021-01-15T11:40:09.303Z
Comments.Text: If the answer below answers your original question (I don't know, to be honest), you should click the "check" symbol and accept it, then, please, flag this comment as "no longer needed", so that it's deleted.
Comments.UserDisplayName: nbro
Tags: <comparison><swarm-intelligence><ant-colony-optimization>
Title: What is the difference between the ant system and the max-min ant system?
UserDisplayName: Pablo
ThreadText: # What is the difference between the ant system and the max-min ant system?
  
  (Asked by: username_0 on 2019-11-02)
  
  I'm studying ant colony optimization. I'm trying to understand the difference between the ant system (AS) and the max-min ant system (MMAS) approaches. As far as I found out, the main difference between these 2 is that in AS the pheromone trail is updated after all ants have finished the tour (it means all ants participate in this update), but in MMAS, only the best ant updates this value. Am I right? Is there any other significant difference?
  
  **Tags:** comparison, swarm-intelligence, ant-colony-optimization
  
  > **Comments:**
  > **username_1**: If the answer below answers your original question (I don't know, to be honest), you should click the "check" symbol and accept it, then, please, flag this comment as "no longer needed", so that it's deleted.  
  
  ---
  
  ## 1 answers
  
  ### Answer by username_2 on 2020-02-08. Score: 2
  
  Let's start by making clear that both AS and MMAS use only global pheromone update. Now, the MMAS has two main differences regarding AS:
  
  1. In AS, all ants that completed a solution are used for the update, while in MMAS only the best ant with a complete solution is used for update (as you had pointed out).
  2. In AS, the pheromone values are not explicitly bounded. In MMAS, the pheromones are enforced to lie within a pre-set interval $\tau\_{min} \leq \tau\_{ij} \leq \tau\_{max}$ (which gives its name to the algorithm). To ensure this condition, the pheromone update is done by means of the formula
  
  $$
  \tau\_{ij} \leftarrow \left[ (1-\rho) \cdot \tau\_{ij} + \Delta \tau\_{ij}^{best} \right]\_{\tau\_{min}}^{\tau\_{max}},
  $$
  
  with the operator $[x]\_a^b$ defined as
  
  $$
  [x]\_a^b=
  \begin{cases}
  a \qquad \mathrm{if} \quad x > a\\
  b \qquad \mathrm{if} \quad x < b\\
  x \qquad \mathrm{otherwise}
  \end{cases}
  $$
  
  Reference: [Ant Colony Optimization. Artificial Ants as a Computational Intelligence Technique](https://courses.cs.ut.ee/all/MTAT.03.238/2011K/uploads/Main/04129846.pdf).
  
  
  ---
  
  