Intelligent Systems II · M2 · formulas

GA and PSO formulas, symbols and memory rules

All M2 formulas

One place for the formulas, meanings, conditions and links to full worked examples.

Binary decoding

x=\sum_{i=1}^{n}b_i2^{n-i}

bᵢ is a bit; the maximum n-bit unsigned integer is 2ⁿ−1.

Full explanation and worked solutions →

Roulette selection

P_i=F_i/\sum_jF_j\quad C_i=\sum_{j=1}^iP_j

Fitness weights must be nonnegative with a positive total. For minimisation use the specified transformation or selection method. C is cumulative probability.

Full explanation and worked solutions →

Population average

\bar f=\frac{f_1+\cdots+f_N}{N}

N is the number of individuals. Compare both best and average; mutation need not improve either.

Full explanation and worked solutions →

PSO velocity

v^{t+1}=wv^t+c_1r_1(pbest^t-x^t)+c_2r_2(gbest^t-x^t)

w weights inertia; c₁r₁ weights personal pull; c₂r₂ weights social pull. Apply the same equation separately to each coordinate.

Full explanation and worked solutions →

PSO position

x^{t+1}=x^t+v^{t+1}

Use the new velocity, not the old one. Then reevaluate the objective.

Full explanation and worked solutions →

Memory updates: minimise → replace a pbest only for a lower score; maximise → only for a higher score. gbest is a position selected from the personal bests. Store the corresponding objective score alongside it. Keep previous best on ties unless instructed otherwise.

Yaad rakho: Formula ke pehle objective direction aur old state likho. Iteration 2 mein fresh positions aur retained memories use karo.

Encoding and minimisation conventions

x=L+\frac{d}{2^n-1}(U-L)

For the specified uniform real-range encoding: decode n bits into unsigned integer d first. L and U are the physical limits. Endpoint examples →

F=\frac{1}{1+f(x)}

This is the stated roulette transformation in our nonnegative-cost minimisation example. It is not required for every GA. It converts a smaller cost into a larger selection weight. Complete minimisation example →