Multi-objective particle swarm optimization with preference information and its application in electric arc furnace steelmaking process

Abstract

In this paper, multi-objective particle swarm optimization with preference information (MOPSO-PI) has been proposed. In the proposed algorithm, the information entropy is employed for measuring the probability distribution of particles; the user’s preference information is represented as the ranking of each particle through the possible matrix. The optimal procedure is guided by the preference information since the global best performance of particle is randomly chosen among non-dominated solutions with higher ranking value in each iteration. Finally, the MOPSO-PI is applied to optimize the steelmaking process; the power supply curve obtained reduces the electric energy consumption, shortens the smelting time and prolongs the lifespan of the furnace lining. The application results show the effectiveness of the proposed algorithm.

Appendix A

1.1 Description of algorithm 1

Each step of proposed MOPSO-PI named algorithm 1 is described as follows.

• 1) Initialize

  The initial population size is Pop, in which each particle has its own position xt and velocity vt. The xi t and vi t are the values of the ith particle at the tth iteration in the update process. The position and velocity for every particle can be specified by Np×Pop matrices, which are initialized randomly within the lower and upper values, where Np is the number of decision variables of the problem. The personal best performance (Pbi) of the ith particle is set to be the position of itself. For each iteration, non-dominated solutions are stored in the external archive, which is also initialized as a null set.

• 2) Convert constrained functions

  For using the multi-objective evolutionary algorithms to solve constrained optimal problem, the constrained problem should be converted into multi-objective problem (Amirjanov 2006). Then, the objective function of constrained problem will be converted into two parts: one part is the original objective function F(x), and the other part is the satisfactory summation function φ(x) under constraint conditions. Therefore, the new objective function E(x) is formulated as follow:

  $$ E\left(\boldsymbol{x}\right)=\left(F\left(\boldsymbol{x}\right),\varphi \left(\boldsymbol{x}\right)\right). $$

  (A1)

  The level of individual x that satisfies the constraint j is φ(x) explained as follows:

  $$ \begin{array}{l}{\varphi}_{g_j}\left(\boldsymbol{x}\right)=\left\{\begin{array}{ll}1,\hfill & {g}_j\left(\boldsymbol{x}\right)<0\hfill \\ {}{g}_j\left(\boldsymbol{x}\right)/{\delta}_j,\hfill & 0\le {g}_j\left(\boldsymbol{x}\right)\le {\delta}_j\hfill \\ {}0,\hfill & otherwise\hfill \end{array}\right.\hfill \\ {}{\varphi}_{h_j}\left(\boldsymbol{x}\right)=\left\{\begin{array}{ll}\left|{h}_j\left(\boldsymbol{x}\right)\right|-{\gamma}_j,\hfill & \left|{h}_j\left(\boldsymbol{x}\right)\right|\le {\gamma}_j\hfill \\ {}0,\hfill & otherwise\hfill \end{array}\right.\hfill \end{array} $$

  (A2)

  In order to find out the feasible solution, the usual methods evaluate every constraint violation value whether small or equal to zero. But we adopt the (Eq. (A2)) to see the satisfactory degree of solution x. In (Eq. (A2)), the parameter δ and γ are the tolerance value, where parameter δ and γ are adapted to reduce the strength of constraints, especially the equality constraints. These two parameters can maintain the diversity of particle population through adding some infeasible individuals.

  At last, the satisfactory summation function φ(x) is defined as follows:

  $$ \varphi \left(\boldsymbol{x}\right)={\displaystyle \sum_{j=1}^l{\varphi}_{g_j}\left(\boldsymbol{x}\right)+{\displaystyle \sum_{j=l+1}^p{\varphi}_{h_j}\left(\boldsymbol{x}\right)}} $$

  (A3)

• 3) Evaluate solutions

  First, for all the solutions, through a rapid dominance sort, the non-dominance particles are saved into the external archive (E _t ), and the dominance particles are discarded out of P _t . After that, if the current population size is not equal to the preset, some new particles randomly generated will add to the current population. Finally, by choosing the Pb _i and the global best performance (Gb) from E _t , the particles are guided by the user’s preference information, which are described in section II.B.

• 4) Update particles

  The velocity matrix and the position matrix are updated according to the following equations:

  $$ {v}_i^{t+1}=\omega {v}_i^t+{c}_1{r}_1^t\left(P{b}_i^t-{x}_i^t\right)+{c}_2{r}_2^t\left(G{b}^t-{x}_i^t\right) $$

  (A4)
  $$ {x}_i^{t+1}={v}_i^{t+1}+{x}_i^t. $$

  (A5)

  where the superscripts t and t + 1 refer to the time index of the current and the next iterations, ω is the inertia weight and decrease according to slope of the current iteration from 0.9 to 0.1. The acceleration coefficients c ₁ and c ₂ are the learning factors of the swarm, which control how far a particle will move in a single iteration, usually c ₁ = c ₂, and value range is [0, 2]. r ₁ and r ₂ are random real values uniformly distributed in the interval [0, 1]. The particles update their velocities and positions by using the current position and velocity information as given in (Eqs. (A4) and (A5)).

• 5) Go back to 3) and termination conditions check

  Go back to 3) and termination conditions check until one of the termination conditions is met. If the termination condition is satisfied, the algorithm terminates and exports the solutions, otherwise, executes sequentially.

  Algorithm 1 Multi-objective Particle Swarm Optimization with preference information

  1) Initialize

  t = 0

  for i = 1 : Pop

  for j =1 : N _p

  x _t  = rand(Min, Max)

  v _t  = rand(Min, Max)

  Evaluate F(x)

  end

  end

  E _t  = []

  2) Convert constraint functions

  Calculate φ(x) for particles

  Generate new objective function E(x)

  3) Evaluate solutions

  t = t + 1

  Sort the particles using quick sort method

  E _t  = E _t ∪ non-dominated solutions

  for i = 1 : N _E

  Evaluate the particles in E _t

  end

  Choose Gb ^t from E _t for the particles

  4) Update particles

  for i = 1 : Pop

  for j = 1 : Np

  Update v ^t and x ^t by (A4) and (A5)

  end

  end

  5) Go back to 3) and termination conditions check

  The parameters used in algorithm are described as follows.

  Rand(Min, Max): Random integer value between Max and Min, v ^t: Velocity matrix of the particles, x ^t: Position matrix of the particles, E _t : External archive, F(x): objective function, φ(x): satisfactory summation function, E(x): new objective function, Pop: Initial population size, N _p : Number of particles, N _E : Number of external archive.