Using multi-population intelligent genetic algorithm to find the pareto-optimal parameters for a nano-particle milling process

Abstract

Nano-particle materials have been widely applied in many industries and the wet-type mechanical milling process is a popular powder technology to produce the nano-particles. Since the milling process involves a number of process parameters and the multi-objective quality criteria, it is very important to set the optimal milling process parameters in order to achieve the desired multiple quality criteria. In this study, a new multi-objective evolutionary algorithm (MOEA), called the multi-population intelligent genetic algorithm (MPIGA), is proposed to find the optimal process parameters for the nano-particle milling process. In the new method, the orthogonal array (OA) experiment is first applied to obtain the analytic data of the milling process. Then the response surface method (RSM) is applied to model the nano-particle milling process and to determine the objective (fitness) value. The generalized Pareto-based scale-independent fitness function (GPSIFF) is then used to evaluate the Pareto solutions. Finally, the MPIGA is proposed to find the Pareto-optimal solutions. The results show that the integrated MPIGA approach can generate the Pareto-optimal solutions for the decision maker to determine the optimal parameters and to achieve the desired product qualities for a nano-particle milling process.

Introduction

Nano-particles are new advanced materials. They refer to particles smaller than 100 nanometers (nm) which may be presented as individual particles or as aggregates. They have been applied in many industrial applications such as photo-catalysis, carbon nanotube, nano-ceramics, fabric fiber, and compound material industries. The wet-type mechanical milling process is a popular powder technology to produce nano-particles because of its applicability to all classes of materials (Frances, 2004, Koch, 1997, Suryanarayana, 2001, Zhang, 2004). The milling process uses stirring and colliding motions in a mill to powder the materials into nanometer size. Since the nano-particle milling process is a complex process and involves a number of process parameters to achieve the desired product qualities, it is very important to optimize the process parameters.

In the wet-type mechanical milling process, the powdered particles have to be dissolved in a liquid to do the impact of collisions in a stirred mill which consists of a rotating agitator and grinding balls media (Mende et al., 2003, Mori et al., 2004, Tsakalakos et al., 2003). The conceptual diagram of a milling process is shown in Fig. 1. The milling process uses the impact of collisions between milling balls media and particles to produce nano-particles. Fig. 2 shows six examples of impact of collisions between milling balls media and particles which might occur during the mechanical milling process. When a mechanical milling process is applied to produce nano-particles, the powdered particles are fractured into smaller particles which have nanometer size. In this research, the wet-type mechanical milling process is used to make titanium oxide (TiO₂) nano-particles. First, the TiO₂ particles are mixed with glycol solvent to form a colloidal solution. Second, a surfactant (phosphate) and the grinding ball media, Zircon dioxide (ZrO₂), are put into the colloidal solution. Third, the required quality criteria (mean of grain size and variance of grain size) of the TiO₂ particles are measured before the colloidal solution is poured into the milling mill. Then the required quality criteria are measured again for the TiO₂ nano-particles after doing the milling process some hours later. Finally, the TiO₂ nano-particles with nanometer size are produced by separating the solute (TiO₂) from the solvent (glycol) in a centrifuge machine. In this research, the Coulter Multisizer machine is used to measure the output quality criteria of nano-particles.

The process parameters that may have significant effects on the required qualities of the nano-particle milling process include the milling time, the flow velocity of circulating system, rotation velocity of agitator shaft, solute-to-solvent weight ratio, and the number of grinding balls. The required qualities of the wet-type milling process of making the nano-particles are that the mean of grain size and the variance of nano-particle grain size must be kept small. Therefore, these two output quality criteria results in a two-objective (Min–Min) optimization problem. However, these two required quality criteria are conflict. If optimizing the process parameters with respect to only one quality criterion, it is possible to obtain a result that is against the other quality criterion. Therefore, it is necessary to capture a set of Pareto-optimal (non-dominated) solutions for the decision maker to determine the optimal parameters in the nano-particle milling process.

Recently, with the rapid development of computational techniques, many multi-objective evolutionary algorithms (MOEAs) have received considerable attention and been recognized to be well suited for solving the multi-objective optimization problems (MOOPs). They can provide all the possible Pareto-optimal solutions to the decision maker (Veldhuzin & Lamont, 2000). The MOEAs are developed based on the two concepts called Pareto-dominance and niching. The Pareto-dominance is used to explore the search space in the direction of the Pareto front and the niching technique is applied to explore the search space along the front to reach diversity. Many well-known MOEA algorithms have been proposed. Schaffer (1985) proposed the vector evaluated genetic algorithm (VEGA). The non-dominated sorting genetic algorithm (NSGA) was first suggested by Goldberg (1989) and then implemented by Srinivas and Deb (1994). Fonseca and Fleming (1993) proposed the multiple objective genetic algorithm (MOGA). Horn (1994) proposed the niched-pareto genetic algorithm (NPGA). Deb, Pratap, Agarwal, and Meyarivan, 2002 proposed the non-dominated sorting genetic algorithm II (NSGA-II). Cochran, Horng, and Fowler, 2003 proposed the multi-population genetic algorithm (MPGA) in solving multi-objective scheduling problems for parallel machines by improving and combining the VEGA and the MOGA methods.

The most important differences among these popular MOEAs are the strategies used to generate the Pareto-optimal solutions. An appropriate design of the fitness evaluation method and the strategies used for generating the Pareto-optimal solutions will result in a good performance of MOEAs (Burke and Silva Landa, 2006, Tan et al., 2006). In the VEGA approach, a population is divided into disjoint sub-populations and each sub-population is assigned to optimize its unique objective function. Each sub-population then evolves separately and an elitist strategy is applied to preserve the best individuals of each objective. The main advantage of the VEGA is that it preserves the best solutions which have the best individual objective values. In addition, it is not difficult to be implemented. Only minor revisions are required to transform the traditional single objective genetic algorithm to VEGA. However, the VEGA approach generates the optimal solution only for each objective function. It does not do a comparison to determine whether one solution dominates the other solution or not. Ho, Shu, and Chen (2004) used a generalized Pareto-based scale-independent fitness function (GPSIFF) that considered the quantitative fitness values in a Pareto space for both dominated and non-dominated individuals to determine the Pareto-optimal solutions. The GPSIFF maintains the essence of domination proposed by Deb (2001), and also has the characteristic of simplicity, generality and effectiveness.

The objective of this research is to propose an integrated approach, called multi-population intelligent genetic algorithm (MPIGA), to find the optimal process parameters for a nano-particle milling process that involves multiple quality criteria to be optimized simultaneously. The MPIGA approach consists of a response surface method (RSM), GPSIFF and VEGA. The parameter design of Taguchi method is applied to economically obtain the output quality criteria measurements and the RSM is used to build the relationship between the process parameters and output quality criteria. The VEGA concept is used to find the single optimized objective, the GPSIFF is used to evaluate the Pareto-optimal solution, and the integrated MPIGA is used to generate the Pareto-optimal solution for a MOOP. In the following sections, RSM and GPSIFF Pareto-optimal solutions are briefly described in Section 2. The proposed integrated approach, MPIGA, is then presented in Section 3. Implementation results of the proposed approach are then illustrated in Section 4. Finally, concluding remarks are made in Section 5.