Max-min ant system (MMAS)

The max-min ant system was introduced by Stützle and Hoos (1997, 1998 and 2000). Its concept is based on using elitism to introduce exploitation to the original ant system and to avoid early stagnation of the search. Stagnation will be described in this section. The MMAS was initially applied to the TSP and QAP and enjoyed greater success than the original Ant System algorithm (Stützle, 1997, Stützle and Hoos, 1998). The MMAS differs in three key aspects from the AS:

(i) Only one single ant is allowed to reinforce pheromone trails after each iteration. This ant may be the one which found the best solution in the current iteration (iteration-best ant) or the one which found the best solution from the beginning of the trail (global-best ant).

(ii) Values for pheromone trails are limited to an interval [τmin,τmax] in order to avoid early stagnation of the search, hence the name max-min.

(iii) Deliberately initialising the pheromone trails ₀ to _max in this way achieves a higher exploration of solutions at the start of the algorithm.

(iv) Pheromone trails are reinitialized when the system approaches are stagnated or when no improved tour has been generated for a certain number of consecutive iterations.

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In MMAS, only one ant is permitted to update the pheromone matrix after every iteration.

This ant provides either the global best ant (gb) or the local best ant (lb) solution. The modified pheromone trail update rule is given by equation (5.11).

_ij(t1)(t)^best_ij (t) (5.11) where

) ( ) 1 (

best ijbest

s t  f

 and s_best may be either the ant with the iteration-best tour or the one

with the global-best tour and  is the pheromone evaporation rate in order to avoid unlimited accumulation of the trail; the value of  should be (0 < ≤1).

The use of only one solution, either the global best (gb) or local best ant (lb), for the pheromone update is the most important means of search exploration in the MMAS. With this choice, solution elements which frequently occur in the best found solutions receive a large reinforcement. In the MMAS, when using only the global-best ant, the search may concentrate too quickly around this early global-best solution and the exploration of possibly better tours is limited. Consequently, the danger of getting trapped in poor quality solutions is higher. On the other hand, using the local best ant favours the exploration of possibly better tours since, especially in the starting phase of the algorithm, the local best ant will differ considerably and early mistakes are more easily avoided. The local and global search is also used in PSO algorithm. The PSO actually uses both aspects of cooperation and competition among the individuals in the population, which means it combines local and global search to reach the global optima, see section 4.3.8. In addition, intermediate approaches can be applied, such as choosing by default the local best ant to update the pheromone trails and using the global best tour only every fixed number of iterations. This mixed strategy which is based on local-best ant and global-best ant for updating pheromone trails will be used in the AOIS problem. The aim of this strategy is to obtain stronger exploration of the search space early in the search and stronger exploitation of the overall best solution later in the run.

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The MMAS and ACS both exploit the best solutions by using only a single ant in the pheromone trail update. However, an important difference between them is the different interpretation of exploitation and exploration. Exploitation in the ACS is mainly interpreted as choosing a high parameter value for

q0, see equation (5.7). In this way, the accumulated knowledge on the problem is exploited by constructing a solution that can be interpreted as a slight modification of the best solution found so far. Exploration in the ACS is obtained using a biased random move according to equation (5.1) with a probability of (1 q ₀) (Stützle, 1998b).

On the other hand, in MMAS, exploitation is mainly interpreted as choosing one single ant, either the local-best ant or the global-best ant, for the pheromone update. Jointly, with a rather high parameter value for ρ, this will slowly shift the probability distribution given by equation (5.1) towards solution components (arcs) which have been shown to be contained in the best solutions. Exploration in the MMAS is derived from explicit pheromone trail limits which aim to increase the solution exploration of the algorithm.

One of the major important features in the MMAS is avoiding of stagnation. Stagnation is the situation in which all ants follow the same path and construct the same tour, which in general is highly suboptimal (Dorigo et al., 1996). In other words, stagnation of the search occurs, for example, in the following situation. If the amount of pheromone on only one arc incident from a node is very high compared to the other arcs, this arc has a high probability of always being selected using equation (5.1). If such a situation occurs at all nodes, the tour corresponding to the best one found so far will be constructed by most ants and the search for better solutions stagnates. To avoid this situation, one possibility is to limit the pheromone trail. This goal can be achieved by limiting the values for pheromone trails to an interval

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[τ_min,τmax]. After updating pheromone at the end of each iteration, the pheromone trails _ijon all arcs are reinforced to be within these limits, as shown in equation (5.12).

^min _ij⁽t⁾^max problem. Clearly, the optimal solution value is not known before the run, and f (sopt) is used as an estimate of that value and then is adapted during the running of the algorithm.

The lower limit of the trail intensity shown in equation (5.14) is calculated with consideration of a number of assumptions. First, it was assumed that the best tours would be found just before stagnation and that, more importantly, better tours were to be found near to the best tours. In such a case, the probability that the best tour found is constructed in one iteration is significantly higher than zero. Through experimentation this property has been shown to be reasonable for TSP benchmark problems (Stützle and Hoos, 2000, Stützle, 1998b). The other assumption was that the main influence on tour construction was the relationship between the upper and lower trail limits.

where _minrepresents the lower limit for the pheromone trail strength and avg is the average number of available options the ant has to choose from at any decision point. The best solution found is constructed with a probability pbest which is significantly higher than 0. The optimal value for probability pbest is 0.05 (Stützle and Hoos, 2000, Ridge, 2007). Other

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authors, such as Ridge (2007), calculated the lower trail limit

n which the pheromone trails are initialised. The pheromone trail reduces due to evaporation to

) limit, because only the best ant is allowed to update the pheromone trail. Arcs which do not obtain regular reinforcement to their pheromone trail will be maintained lower and be chosen more rarely by the ants. In this sense, errors made in the past are avoided in the MMAS. An error is associated with choosing arcs that lead to fairly bad tours; these are denoted as poor arcs. Thus, the pheromone trail on poor arcs decreases gradually and only good arcs keep a higher level of pheromone. These good arcs are then combined by the probabilistic tour creation to generate improved tours (Stützle and Hoos, 2000).

Stützle and Dorigo (1999) applied the MMAS to the TSP and their results compared to Iterated Local Search. The MMAS found a better solution than the other algorithms used for 77% of the problems on an average run. The MMAS was used in Stützle (1998b) to attack the Flow Shop Problem. The algorithm outperformed a number of other methods such as Simulated Annealing and Multiple Descent. The experimental results in Stützle and Hoos (2000) demonstrate that the MMAS achieves strongly improved performance compared to the AS and to other improved versions of the AS for the TSP; moreover, the MMAS is among the best available algorithms for the QAP. The MMAS was applied to the University Timetabling Problem by Socha et al. (2002 and 2003). The paper showed that the algorithm performed better at a set of problem instances than an algorithm using the local search with

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random starting solutions. Ning et al. (2010) applied the MMAS to the construction site layout planning (CSLP) problem. CSLP problem is a dynamic multi-objective optimisation problem as there are different facilities employed in the different phases of a construction project. The CSLP problem was modelled as a QAP under two objective functions of minimising the representative score of safety/environmental concerns and the total handling cost of interaction flows between the facilities associated with the construction site layout.

The experimental results show that the safety level is improved and the construction cost is reduced.

 Features of MMAS algorithm

The following are the important features for the MMAS algorithm:

1. One of the major important features in the MMAS algorithm is avoiding too early stagnation.

2. By using only one single ant in the MMAS algorithm the pheromone trail update, the best solutions can be better exploited.

3. By using the adaptive memory allows the previously visited workstations to be kept by means of the pheromone trail matrix, which is used to influence the construction of new better solutions.

4. The heuristic information helps to find acceptable solution in the early stages of the search process.

5. The collective interaction of a population of ants leads to increase the exploration of the search space.

6. MMAS algorithm starts with initial high pheromone trial which is leads to increase exploration of the search space.

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