Como inciden las prácticas de RSE de las empresas sobre las influencias

Based on the problem modeling, we adopt one of the most successful ACO algorithms labeled as MMAS for tackling the foraging problem. As shown in [27], the MMAS typically has three phases: 1) construct ant solutions, 2) apply local search (optional), and 3) update pheromone values. The realization of these three phases is problem-specific. Considering the target optimal execution order search problem, the MMAS is tailored into an iterative procedure to identify the optimal execution order, see the block diagram in Fig. 6.3. In the following, we explain the functional units in Fig. 6.3 one-by-one.

Ant Trial Construction

We adopt M ants. Based on the up-to-date distribution of pheromone trails in the tree graph, one ant selects the up-to-date dominant path. The others stochastically select paths. Dominant Path Identification Starting from the root node, whenever a step node with out-degree two is reached, the branch with the higher pheromone value is selected to pro- ceed. Until a leaf node is reached, the traversed path is the so-called dominant path. After the iterative training process terminates, the execution order associated to the dominant path is considered to be the solution to the optimal execution order search problem. Stochastic Path Selection An ant starts its trip from the root node. Whenever a new step node is reached, it chooses one of the expanded branches to proceed the travel. If the step node has only one expanded branch, the decision is straightforward. When the step node has out-degree two, it stochastically selects either branch to proceed according to the pheromone values on both branches. When a leaf node is reached, the traveled path is considered to be selected by the ant. Each ant makes its own decision independently.

Specifically, the stochastic decision of ants made at each out-degree-two step node is realized as follows. Assume the antm reaches a step node o with two expanded branches representing the inner and outer iteration, i.e.,B1,oandB2,o. The probability that the ant

m chooses Bk,owithk ∈ {1, 2} is set as

pk,o=

τk,o

τ1,o+ τ2,o

(6.2) whereτk,ois the up-to-date pheromone value on the branchBk,o. BetweenB1,o andB2,o,

the branch with higher pheromone value is more likely to be chosen than the other. The pheromone valuesτ1,oandτ2,o are determined based on the trial quality of ants. For real-

izing stochastic selection, we employ a random number generator, which uniformly gen- erates a random number within the region [0, 1]. If the generated number falls into the interval [0, p1,o], the ant m will proceed to the next step node through the branch B1,o;

otherwise, the antm takes the branch B2,o.

Training Frame Generation

The information bit sequence is uniformly sampled from_{{0, 1}}Nm _{followed by encoding}

and modulation according to the given transmission strategy. With respect to the given channel statistics, the fading channel coefficients and also the additive white Gaussian noise are generated accordingly. Each training frame is independently formed.

Trial Quality Evaluation

For evaluating the trial quality of the M ants, two aspects are taken into account. One is whether an ant can find a food source along the selected path. We count the number of training frames per training iteration that each ant fails to find the food source, where the number associated to the antm is denoted as Nfail(m). Travel cost is the other aspect.

Denoting the travel cost of the ant m as C(m), it equals the cumulative computational energy consumption of the execution order associated to the path selected by the antm. Condition Check

The number of training frames that are generated within each training iteration should be chosen to yield_{Nfail(m), C(m)} such that a reliable comparison among the M ants can be

made. Let us denote the ant that selects the up-to-date dominant path asm[dom]_{. Based on}

Nfail(m[dom]), we here propose to terminate each training iteration if Nfail(m[dom]) is larger

than a pre-defined threshold, i.e.,Nthr.

Pheromone Update Rule

Taking the branchBk,oas an example, the standard pheromone update rule adopted in the

MMAS [27, 111] is given as

whereρ ∈ (0, 1) is the evaporation rate; ∆τk,o is the pheromone adjustment that can be

an increment or a decrement; and the parameterτmax andτmin are used to limit the value

ofτk,o within the region[τmin, τmax]. Following the guidelines in [26, 82], the parameters

{ρ, τmax, τmin} in (6.3) can be determined based on the number of training iterations Niter

and the number of antsM ρ = 1− Niter s 1₋ M√ 0.05 (M₂ − 1) · M√ 0.05, τmin = τmax· (1 − M √ 0.05) (M₂ − 1) · M√ 0.05 . (6.4)

Let us normalizeτminto one and initialize the pheromone value on each branch withτmin.

For the setting of the pheromone adjustment ∆τk,o in MMAS, there is no fixed rule

in the literature. It needs to be tuned on the considered problem. Typically, it is ob- tained in an empirical way. Here, we propose to determine∆τk,oas follows. According to

{Nfail(m), C(m)}, we identify the good and also the bad ones among the M ants. Accord-

ing to the goal of scheduling, a good ant should achieve a smallNfail(m) at the cost of a

smallC(m). In equations, the good ants belong to Qgd ∆ =  m    m∈ Q ′_{, C(m) =}  min m′_∈Q′C(m ′₎  (6.5) where the set of_Q′is defined as

Q′ ∆₌  m   

m ∈ {1, 2, . . . , M}, ζ · Nfail(m)≤ minm′ Nfail(m

′₎



(6.6) with the parameter ζ ∈ (0, 1]. Intuitively, the parameter ζ should equal one, as we are interested in achieving the minimum FER. However, in reality, we often consider that two decoding processes have identical decoding performance as long as the difference between their FERs is inappreciable. For instance, the difference between the FER of0.012 and 0.01 is negligible, although the former one is mathematically larger than the latter one. Given this argument, the use ofζ in (6.6) is to identify some of{Nfail(m)} that can be considered

to have an inappreciable difference with the minimum of_{Nfail(m)}. Here, we exemplarily

let it equal0.86. In contrast to good ants, the set of bad ants is defined as Qbd ∆ =  m   

m ∈ {1, 2, . . . , M}, Nfail(m)C(m) > maxm′_∈QgdNfail(m

′_{)C (m}′₎



. (6.7)

Based on_Qgd and_Qbd, the pheromone adjustment is determined as follows: • If the branch Bk,ois visited by any good antm[gd] ∈ Qgd, we have

∆τk,o = max

m∈Qgd, m visits Bk,o

ρτmax·

Nmedian+ Nfail m[dom]

Nmedian+ Nfail(m) ·

C m[dom]

C (m) . (6.8)

whereNmedianrepresents the median of{Nfail(m)}. In the above equation, the incre-

ment is generated by comparing the trial quality of good ants with that of the ant selecting the up-to-date dominant path. At initial iterations, the up-to-date domi- nant path can be suboptimal. Namely, the productNfail m[dom]

larger thanNfail m[gd]

C m[gd]
_{. The correction on the concentration of pheromone}

trails is done by adding a large increment onto the path associated to the good ant at the present iteration. As the training process proceeds, we expect the up-to-date dominant path becomes optimal or near optimal. By then, the increment is (approx- imately) equal toρτmax. From (6.3), we note that the pheromone value of any branch

is always upper bounded byτmax. As such, the following inequality must hold

(1− ρ)τk,o+ ρτmax = τk,o+ ρ· (τmax− τk,o)≥ τk,o (6.9)

which means whenever a branch is visited by a good ant its pheromone value after the update shall be greater than or equal to the value before the update. It is fur- ther noted that if a branch is visited by more than one good ant we take the largest possible increment on that branch.

• If the branch Bk,ois only visited by a bad antm[bd] ∈ Qbd, we have

∆τk,o= max m∈Qbd, m visits Bk,o−ρτ max·  1− maxm′∈QgdNfail(m ′_)C(m′₎ Nfail(m)C(m)  . (6.10) The decrement is particularly large if the trial quality of the bad ant is much worse than that of the good ants.

• For the branches not covered by the previous cases, we have ∆τk,o = 0.

In addition to the above-described pheromone update rule, we further introduce a post- pheromone adjustment for the situation in which the two branches expanded from the same step node have their pheromone values equal to the up limitτmax. In order to identify

which one of them is superior in the next training iteration, we simultaneously reduce the pheromone values on both branches to(τmax+ τmin)/2.