# Extension Of Max-Min Ant System With Exponential Pheromone Deposition Rule

Ayan Acharya \#1, Deepyaman Maiti\#2, Aritra Banerjee\#3, R. Janarthanan*4**, Amit Konar** \#5
\# **Department of Electronics and Telecommunication Engineering, Jadavpur University** 
Kolkata: 700032, India 1 masterayan@gmail.com 2 deepyamanmaiti@gmail.com 3 aritraetce@gmail.com 5 konaramit@yahoo.co.in 
* **Jaya Engineering College** 
C.T.H.Road, Thiruninravur, Chennai 602024, Thiruvallur District, Tamilnadu, India 4 srmjana_73@yahoo.com Abstract**— The paper presents an exponential pheromone** 
deposition approach to improve the performance of classical Ant System algorithm which employs uniform deposition rule. A simplified analysis using differential equations is carried out to study the stability of basic ant system dynamics with both exponential and constant deposition rules. A roadmap of connected cities, where the shortest path between two specified cities are to be found out, is taken as a platform to compare MaxMin Ant System model (an improved and popular model of Ant System algorithm ) with exponential and constant deposition rules. Extensive simulations are performed to find the best parameter settings for non-uniform deposition approach and experiments with these parameter settings revealed that the above approach outstripped the traditional one by a large extent in terms of both solution quality and convergence time. 

## I. I**Ntroduction**

 Stigmergy **is a special kind of communication prevalent** among many species of ants. While roaming from food sources to the nest and vice versa, ants deposit on the ground a substance called pheromone, forming in this way a pheromone trail. Ants can smell pheromone and choose, in probability, paths marked by stronger pheromone concentration. Thus the pheromone trail allows the ants to find their way back to the food source or to the nest. Denebourg et al. [4] first studied the pheromone laying and following behavior of ants. Ant System (AS) and Ant Colony Optimization (ACO) actually owe their inspiration to the works of Denebourg et al. The paper attempts to extend the ant system model by introducing an exponential pheromone deposition approach. We solve the deterministic ant system dynamics using differential equation. The analysis helps in determining the range of parameters in both forms of pheromone deposition rule to confirm stability in pheromone trails. The deterministic solution undertaken does not violate the stochastic nature of the Ant System algorithm since a segment of trajectory here is also selected probabilistically. 

 The apparent correlation between the selection of exponential pheromone deposition approach and the expected improved convergence time as well as solution quality of extended AS can be explained in the following way. A uniform pheromone deposition by an ant cannot ensure subsequent ants to follow the same trajectory. However, an exponentially increasing time function ensures that subsequent ants close enough to a previously selected trial solution will follow the trajectory, as it can examine gradually thicker deposition of pheromones over the trajectory. Naturally, deception probability **([3]) being less, convergence time** should improve. Our previous work [8] was based on stability analysis using difference equation. In this paper, we have employed differential equations which not only characterize the system dynamics more precisely but also are more popular than difference equation. The previous paper lacked sufficient experimentations to establish the betterment of the proposed deposition rule. The experiments performed over TSP
instances could not at all highlight the philosophy of the non uniform deposition rule. This paper presents sufficient simulation results to establish the proposed algorithm's superiority over the traditional one. Problem environment is also chosen very cleverly to emphasize the efficacy of the proposed algorithm. Exhaustive experimentations also help find out the suitable values of parameter for which the proposed algorithm works best and from these results we attempt to ascertain an algebraic relationship between the parameter set of the algorithm and feature set of the problem environment. 

 The paper is divided into seven sections. Section II gives a brief description of AS (Ant System) and MMAS (**MAX-MIN** Ant System). **In section III, a scheme for the general solution** of Ant System is formulated. Stability analysis with closed form solution of different pheromone deposition rules is undertaken in section IV. Parameter settings for MMAS are provided in a separate module in section V. Performance 