3.1 SSA
The hunt procedure starts when flying squirrels begin scavenging [26]. During fall, squirrels look for nourishment assets by skimming from one tree to the next. At the same time, they change their areas and investigate various regions of woods. As the climatic conditions are sufficiently hot, they can meet their every day vitality needs more rapidly on the eating routine of oak seeds accessible in bounty and thus, they devour oak seeds quickly after discovering them. Subsequent to satisfying their day by day vitality prerequisite, squirrels scan for ideal nourishment hotspot for winter (hickory nuts). Capacity of hickory nuts will help them in keeping up their vitality prerequisites in harsh climate, decrease the expensive searching excursions and increase the likelihood of endurance.
During winter, lost leaves spread in deciduous woodlands result an expanded danger of predation and thus, squirrels become less dynamic but still remain active. Toward the finish of winter season, squirrels again become dynamic. This is monotonous procedure and structures the establishment of SSA. The SSA approach refreshes the places of squirrels as indicated by the ebb and flow season, the sort of squirrels and if chasers showing up.
3.1.1 Instate the population
Assuming the number of squirrels is N, and the upper and lower limits of the pursuit space are XU and XL, the N squirrels are arbitrarily created as:
$$ {X}_i={X}_L+\mathit{\operatorname{rand}}\left(1,D\right)\times \left({X}_U-{X}_L\right) $$
(12)
where Xi indicates the ith squirrel, (i = 1: N), rand () is an random number in the range of 0 and 1, and D is the measurement of the issue.
3.1.2 Group the population
SSA requires that there is only a single squirrel at each tree, so for N squirrels, there are N trees in the woods. Among the N trees, there is one hickory tree and Na oak seed trees, while the rests are typical trees having no nourishment. The hickory tree is the best nourishment asset for the squirrels while the oak seed trees come the second. Positioning the fitness estimations of the populace in rising request, the squirrels are separated into three kinds:
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Squirrels situated at hickory tree (Wh);
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Squirrels situated at oak seed trees (Wa);
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Squirrels situated at ordinary trees (Wn).
3.1.3 Refresh the location of squirrels
The squirrels refresh their situations by skimming to the hickory tree or oak seed trees as follows:
$$ {X}_i^{t+1}=\left\{\begin{array}{c}{X}_i^t+{d}_g{G}_c\left({X}_{ai}^t-{X}_i^t\right)\kern1em if\ {r}_2\ge {P}_{dp}\\ {} Random\ location\kern2.00em \ otherwise\end{array}\right. $$
(13)
$$ {X}_i^{t+1}=\left\{\begin{array}{c}{x}_i^t+{d}_g{G}_c\left({X}_h^t-{X}_i^t\right)\kern0.5em if\ {r}_3\ge {P}_{dp}\\ {} Random\ location,\kern0.5em otherwise\end{array}\right. $$
(14)
Pdp is esteemed at 0.1 and indicates the chaser likelihood. In the event that r > Pdp, no chaser shows up, and the squirrels coast in the backwoods to discover the nourishment and are protected. If r < Pdp, the chasers show up, and the squirrels are compelled to limit the extent of exercises and are imperiled, and their locations are migrated arbitrarily. dg is the skimming separation that can be determined by:
$$ {d}_g=\frac{h_g}{\mathit{\tan}\left(\varphi \right)} $$
(15)
where hg is the constant estimated 8, tan (ɸ) indicates the coasting point that can be determined by:
$$ \mathit{\tan}\left(\varphi \right)=\frac{D}{L} $$
(16)
The drag power and lift power can be estimated as:
$$ D=\frac{1}{2\rho {V}^2S{C}_D} $$
(17)
$$ L=\frac{1}{2\rho {V}^2S{C}_L} $$
(18)
3.1.4 Occasional changeover verdict and arbitrary refreshing
Toward the start of every generation, SSA necessitates that the entire populace is in winter, which implies that the locations of all squirrels are updated by (11) and (12). At the point when the squirrels are refreshed, regardless the season, change is decided by the following formulae:
$$ {S}_c^t=\sqrt{\sum_{k=1}^d{\left({X}_{ai,k}^t-{X}_{h,k}^t\right)}^2}i=1,2,..,{N}_a $$
(19)
$$ {S}_{min}=\frac{10{e}^{-6}}{365^{t/\left({t}_{max}/2.5\right)}} $$
(20)
If Stc < Smin, winter is finished and the season goes to summer, otherwise the season is unaltered. At the point when the season goes to summer, the squirrels who float to Wh remain at the refreshed area, while the squirrels skimming to Wa and not meeting with chasers move their situations as follows:
$$ {X}_{inew}^{t+1}={X}_L+ Le^{\prime } vy(x)\times \left({X}_U-{X}_L\right) $$
(21)
Le’vy is the arbitrary walk model whose progression complies with the Le’vy appropriation and can be determined by:
$$ L{e}^{\prime } vy(x)=0.01\times \frac{\alpha \times {r}_a}{{\left|{r}_b\right|}^{\frac{1}{\beta }}} $$
(22)
where α is determined as:
$$ \kern1em \alpha ={\left[\frac{\varGamma \left(1+\beta \right)\times \sin \left(\frac{\pi \beta}{2}\right)}{\varGamma \left(\frac{1+\beta }{2}\right)\times \beta \times {2}^{\left(\frac{\beta -1}{2}\right)}}\right]}^{\frac{1}{\beta }} $$
(23)
3.2 Fuzzy decision strategy
The objective functions, viz. economic and emission dispatch shall be used in parallel in the multi-objective EELD problem. However, this makes the comparison of the two solutions difficult. Neither solution vector X1 nor X2 can be superior to each other if they are Pareto optimal, because if superior result is obtained from X1 for an objective, X2 would offer improved performance for another objective. Obtaining the best solution from multiple non-dominated solutions is challenging in multi-objective EELD problem, though it is always possible to collate these outcomes and obtain the best compromised solution. For achieving this, one has to use the proper mechanism to combine both objectives and ensure that it conforms to the target and preference of the decision maker.
Researchers commonly use fuzzy set theory to arrive at the best solution amongst many uncontrolled solutions. It is implausible to achieve both least fuel cost along with least emission as they are contrary to each other. But it is feasible to build a dispatch option that can optimize both. Fuzzy membership functions assign Degree of agreement (DA) to each objective, and merit of the objective is reflected by DA in a linear scale of 0 – 1 (worst to best). Fj is a solution in the Pareto-optimal set in the jth objective function and is defined by a membership function as:
$$ \mu \left({F}_j\right)=\left\{\begin{array}{c}1\kern6.75em if\ {F}_j\le {F}_j^{min}\\ {}\frac{F_j^{max}-{F}_j}{F_j^{max}-{F}_j^{min}}\kern1em if\ {F}_j^{min}\le {F}_j\le {F}_j^{max}\\ {}0\kern6.75em if\ {F}_j\ge {F}_j^{max}\end{array}\right. $$
(24)
For each non-dominated solution, the normalized membership function \( {\mu}_D^k \) can be calculated as:
$$ {\mu}_D^k=\frac{\sum_{i=1}^2\mu\ \left({F}_i^k\right)}{\sum_{k=1}^M{\sum}_{i=1}^2\mu\ \left({F}_i^k\right)} $$
(25)
The solution that contains the maximum of \( {\upmu}_{\mathrm{D}}^{\mathrm{k}} \) \( {\mu}_D^k \) based on cardinal priority ranking is the best compromised solution, i.e.:
$$ \mathit{\operatorname{Max}}\left\{{\mu}_D^k:k=1,2,..,M\right\} $$
(26)
