*2.4. Application of HBMO Algorithm for Reservoir Rule Curves Generation* 2.4.1. HBMO Algorithm

The HBMO Algorithm is a hybrid search algorithm based on bee mating behavior. The biological bee breeding process is transformed into a mathematical modeling program. As a result, the phases in the adjustment process were properly outlined. Mating is the first step in algorithm development. Every queen bee makes a flight based on her power and speed throughout each mating flight. Equation (8) determines the likelihood of mating between individual male bees and queen bees. The likelihood of mating is high during the start of the mating flight when the queen bee's velocity is high, or when a male bee is sufficiently numerous to mate, the probability of mating is high.

After the movement of the queen bees or after mating, energy, and speed decrease according to Equations (9) and (10). When all queen bees have completed a pairing flight, they begin to breed to achieve the required number of embryos. The queen bees are selected in proportion to the queen bee's fitness and are artificially inseminated with sperm randomly selected from the queen bee's sperm sac. The worker bees would be selected in proportion to their fitness to be used to improve larval outcomes. After the embryos were born, they would be sorted according to their fitness. The best larvae replace the worst queen bees until there are no better embryos than any queen bees. The remaining larvae are then killed and new matings begin until there is a perfect mating. All predetermined will be completed or meet converging criteria [42].

$$Prob\left(Q, D\right) = e^{-\frac{\Lambda(f)}{S(t)}}\tag{8}$$

where *Prob* (*Q*, *D*) is the probability of mating between the male bee *D* and the queen bee *Q* or the probability of successful mating; Δ(*f*) is the difference between the male bee's fitness (*f*(*D*)) and the fitness of the queen bee (*f*(*Q*)); *S*(*t*) is the speed of the queen bee at the time.

$$E(t+1) = E(t) - \gamma \tag{9}$$

$$S(t+1) = \mathfrak{a} \times S(t) \tag{10}$$

where *E*(*t*) is the queen's energy; *S*(*t*) is the queen's speed; *α* is a factor ∈ [0, 1] and *γ* is the amount of energy reduction after each transition.
