# What is the difference between the ant system and the max-min ant system?

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?

• 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.
– nbro
Jan 15 at 11:40

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
$$[x]_a^b= \begin{cases} a \qquad \mathrm{if} \quad x > a\\ b \qquad \mathrm{if} \quad x < b\\ x \qquad \mathrm{otherwise} \end{cases}$$