*4.4. Employed Bees*

During the employed bee phase, each individual tries to search around the food source to obtain a better food source. The food source is the solution to the problem.

In order to allow the employed bee to fully explore around the solution, a local search strategy based on ranking is proposed, with the central idea that high-quality solutions are used to guide bad solutions to update themselves.

First, there is a requirement to identify high quality individuals in population. A new way of determining high-quality individuals is proposed. The quality of each individual is related to two factors: the number of dominant solutions and the similarity to the ideal solution. (21) gives a high-quality assessment function for each individual, where *ni* denotes the number of solutions in population that are dominated by the current individual *i*, *d+*, and *d–* denote the Euclidean distances to the ideal and negative ideal solutions, respectively. (22) gives the formula for the Euclidean distance, where *xi* denotes the *i*-th subproblem of the current solution and *x\*i* denotes the *i*-th subproblem of the ideal solution. Since this paper is about finding a minimum of two objectives, the ideal solution is the lower boundary of the search space and the negative ideal solution is the upper boundary of the search space.

$$value\_i = \frac{d\_i^-}{d\_i^- + d\_i^+} + \frac{N - n\_i}{N^2} \tag{20}$$

$$d = \sqrt{\left(\sum (\mathbf{x}\_i - \mathbf{x}\_i^\*)^2\right)}\tag{21}$$

The high-quality individuals then guide the poor individuals to self-renewal when the employed bees search around solutions. The high-quality individuals guided the poor individuals to different degrees, and (23) gives the degree to which each individual *i* guided the poor individuals. It is worth noting that the high-quality individuals only guide the poorer individuals in their neighborhood. The Euclidean distance of each individual *i* in population from other individuals was calculated and the nearest *T* individuals were selected as neighbors of *i*.

$$d = \pi \ast \frac{n\_i}{N} \tag{22}$$

In addition, in order to prevent individuals in the population from leading differential updates that affect other individuals that have already been updated and destroy the structure of individuals, individuals in population are sorted in a non-ascending order according to their quality, and individuals that have already been updated do not participate in updates in the same population.

It is also worth noting that five update strategies are used in this paper, depending on the problem to be solved. These strategies are insertion and exchange of working sequences, mutation of velocity matrices, and insertion mutation and cross mutation of working sequences and velocity matrices. The employed bees obtain possible solutions based on these update strategies.

The employed bees search around the solution starting from the first update strategy. If the currently selected update strategy does not yield a solution with high fitness, then the next employed bee searches based on the next update strategy until it finds a high-quality solution. When all five update strategies have been searched, the search starts from the first one again. The flow of the employed bee phase is shown in Algorithm 3, where *Quality*() means calculating the quality of each individual according to (21), *Level*() means determining the degree to which an individual leads the difference solution, *GetNew*() means updating individuals according to the strategy *qi* with an initial value of 1 for *qi*, and *GetBad*() means obtaining the difference solution that has a high similarity to the current individual and has not been updated.
