Pheromone

Pheromones are central to the whole movement process within the MPACA. In this mechanism each feature present, irrespective whether ordinal or nominal, has a representative pheromone value. For example a nominal feature like colour could have pheromone indicating “red”, which differs from pheromone indicating “blue”. A similar mechanism applies to normalised ordinal value features: separate pheromones are associated with each feature value on each edge and

ants will match any pheromone for that dimension that occurs within its own range. When ants acquire (learn) new features, they no longer deposit pheromone for a single feature value, but deposit pheromones for combined features together.

The point at which pheromone deposition takes place is a core consideration in ACO litera- ture. Does pheromone deposition take place once all ants traverse the entire graph, or does the pheromone deposition take place at each edge traversal, or is it a combination of both? [see sec- tion (2.3.4)]. In order to keep in check the decentralisation aspect of the algorithm, the MPACA only applies pheromone deposition at the local ant movement level.

3.5.1.1 Ant Deposit State

Ants only lay a pheromone trail when departing nodes which have feature values that match ones being sought by the ant. The ant ceases to deposit pheromone once it arrives at a node where the features present are uninteresting (i.e. do not match). Pheromone deposition is only restarted once the ants reach a new interesting node, outlined in algorithm (5).

Algorithm 5 Pheromone deposition

Comment: Each time increment represents a single step moved by an ant while (MPACA is in running mode) do

if (Ant at a node) then

Ant deposit mode is set to non-deposit

if (Feature set on the node matches ant’s carried features list) then Set ant to deposit mode

end if

Ant chooses an edge in accordance with the edge-selection process Ant executes the first step on the edge

end if

if (Ant is on an edge) then

Ant moves one step forward onto the edge it is traversing if (Ant is in deposit mode) then

Ant deposits a unit of pheromone signalling the features it is carrying end if

end if end while

The MPACA can potentially add a pheromone representation on each connecting edge for each feature value in the system. Thus, the dimensionality of the problem and the dataset size deter- mine the number of pheromone types that can be laid down.

3.5.1.2 Pheromone Quantity Deposited

The MAX-MIN Ant System (MMAS) is one of the more effective ACO implementations, sec- tion (2.3.4), and gains much of its power from the fact that it limits the amount of pheromone

that can be deposited. The upper bound limits the excessive build-up of pheromone on any one edge. It is important to note that if unconstrained, the auto-catalytic effect of pheromone can precipitate stagnation at local maxima. Following from the MMAS, the MPACA uses a simi- lar mechanism that increases exploration and limits the likelihood of stagnation via an inbuilt maximum pheromone value.

Each time pheromone is laid down, the amount deposited is compared against a maximum co- efficient of the standard pheromone deposition value. The upper ceiling is controlled by the maximum coefficient parameter, τmax. τmax serves as a co-efficient of τQ. That is, when τ (t) ≥ (τQ× τmax), than the amount of pheromone deposited is capped and set to τQ× τmax. There is no fitness function in the MPACA which determines the quantity of pheromone that is to be added. Instead, ants deposit pheromone equally on all edges they are traversing for each feature value they are carrying. This makes the MPACA more decentralised than other approaches. The amount of pheromone at a step is expressed by equation (3.3):

τ (t) = τ (t − 1) + (τQ) where τ(t) ≤ τQ× τmax (3.3)

where τ(t) is the amount of pheromone to be deposited at time t, τ(t − 1) is the current amount of pheromone present at this same step t − 1, and τQ is the amount pheromone that can be deposited.

Coupled with the maximum amount of pheromone, there is also an additional minimum tol- erance parameter, τmin, which serves as a pheromone cleansing mechanism. If the amount of pheromone is lower than τmin, then this value is removed from the system. Thus, there are three key considerations for pheromone deposition:

• The “pheromone quantity being deposited”, τQ;

• The “maximum ceiling value” , τmax, or the maximum coefficient parameter; and • The “minimum clearing value”, τminor the minimum tolerance parameter.

3.5.1.3 Pheromone Evaporation

The available literature presents various ways of how pheromone evaporation occurs. In most cases deposition and evaporation are combined into one process. This cannot be done in the MPACA, since pheromone deposition does not occur simultaneously on all edges. In the MPACA, evaporation is applied to each and every pheromone scent present at every given interval in

time→ t, via a percentage reduction expressed by equation (3.4):

τ (t) = [τ (t − 1) × (1 − ρ )] (3.4)

where τ(t) is the amount of pheromone present at a point in time t, τ(t − 1) is the current amount of pheromone present at the same point previously at time t − 1, and ρ is the pheromone evaporation rate.

Since a percentage reduction applied to any value can never reach zero, the algorithm filters out values below τmin. This is a variation from the MMAS, which applies a lower bound on each edge, τmin, where each edge always has a minimum pheromone value.

The quantity of pheromone deposited is interlinked with the size of the graph. The pheromone evaporation parameter controls the amount of pheromone that can exist on any edge. It follows that there must be a balance between the amount of pheromone deposited and the pheromone evaporation rate itself. If the pheromone quantity is Q, and evaporation rate is E, then E should consist of a value per unit time substantially less than Q, otherwise the Q value would be nulli- fied. Ideally, these values are chosen by the operator of the algorithm or extracted empirically via experimentation. Irrespective of the approach, the MPACA cannot at this point self-adjust these values for optimal performance.

If the graph space is overly-compressed, higher pheromone evaporation is required in order to ensure that stagnation does not occur, as smaller evaporation intervals would not manage to decrement enough pheromone and the solution could at some point have equal maximum amounts of pheromone on each edge.