Modified differential evolution (MDE) for optimization of non-linear chemical processes

doi:10.1016/j.compchemeng.2005.12.020

Computers & Chemical Engineering

Volume 30, Issues 6–7, 15 May 2006, Pages 989-1002

https://doi.org/10.1016/j.compchemeng.2005.12.020 Get rights and content

Abstract

In recent years, evolutionary algorithms (EAs) are gaining popularity for finding the optimal solution of non-linear multimodal problems encountered in many engineering disciplines. Differential evolution (DE), one of the evolutionary algorithms, is a novel optimization method capable of handling nondifferentiable, non-linear and multimodal objective functions. Previous studies have shown that differential evolution is an efficient, effective and robust evolutionary optimization method. Still, DE takes large computational time for optimizing the computationally expensive objective functions. And therefore, an attempt to speed up DE is considered necessary. This paper introduces a modification to original DE that enhances the convergence rate without compromising on solution quality. Our modified differential evolution (MDE) algorithm utilizes only one set of population as against two sets in original DE at any given point of time in a generation. Such an improvement reduces the memory and computational efforts. The proposed MDE is applied to benchmark test functions and five non-linear chemical engineering problems. Results obtained are compared with those obtained using DE by considering the convergence history (CPU time and the number of runs converged to global optimum) and the established statistical techniques, taking into account the variability in the results, such as t-test. As compared to DE, MDE is found to perform better in locating the global optimal solution for all the problems considered.

Introduction

Many engineering optimization problems contain multiple optimal solutions, among which one or more may be the absolute minimum or maximum solutions. These absolute optimum solutions are known as global optimal solutions and other optimum solutions are known as local optimal solutions. Ideally, we are interested in the global optimal solutions because they correspond to the absolute optimum objective function value.

Most of the traditional optimization algorithms based on gradient methods have the possibility of getting trapped at local optimum depending upon the degree of non-linearity and initial guess. Unfortunately, none of the traditional algorithms are guaranteed to find the global optimal solution, but population based search algorithms are found to have a better global perspective than the traditional methods (Onwubolu & Babu, 2004). In the recent past, non-traditional search and optimization techniques based on natural phenomenon (evolutionary computation) such as genetic algorithms (GA) (Holland, 1975, Goldberg, 1989), evolution strategies (ESs) (Schwefel, 1981), simulated annealing (SA) (Kirkpatrick et al., 1983), and differential evolution (DE) (Price & Storn, 1997) to name a few, have been developed to overcome the problems. Among their advantages are: (1) they do not require the objective function to be continuous and/or differentiable, (2) they do not require extensive problem formulation (in case of traditional methods such as Integer programming, geometric programming, branch and bound methods, etc., special mathematical formulation is required for solving a problem), (3) they are not sensitive to starting point, (4) they usually do not get stuck into so called local optima, and (5) they are more likely to find out a function's true global optimum. These advantages enhance their application to various fields. They have been successfully applied in many engineering design problems (Androulakis & Venkatasubramanian, 1991; Angira & Babu, 2003; Babu, 2004; Babu and Angira, 2002a, Babu and Angira, 2002b; Babu, Pallavi, & Syed Mubeen, 2005; Babu & Sastry, 1999; Chiou, Chang, & Su, 2004; Chiou & Wang, 1999; Deb, 1996, Deb, 2001; Hendtlass, 2001; Lee, Han, & Chang, 1999; Lu & Wang, 2001; Storn, 1995, etc.) to name a few. Recently, Onwubolu and Babu (2004) compiled new techniques and their applications to various disciplines of engineering and management.

Previous studies have shown that DE is an efficient, effective and robust evolutionary optimization method. Still, DE takes large computational time for optimizing the computationally expensive objective functions. And therefore, an attempt to speed up DE is considered necessary. In this paper, modified differential evolution (MDE), an evolutionary optimization technique, is proposed that utilizes only one set of population as against two sets in original DE at any given point of time in a generation. Further, it is applied to nine benchmark test functions and five non-linear chemical processes. Also the performance of MDE is compared with that of differential evolution (DE). The comparison is made by considering the convergence history (CPU time and the number of runs converged to global optimum) and the established statistical techniques, taking into account the variability in the results, such as t-test.

Section snippets

Differential evolution (DE) in brief

Differential evolution (DE), a recent optimization technique, is an exceptionally simple and easy to use evolution strategy, which is significantly faster and robust at numerical optimization and is more likely to find a function's true global optimum (Price & Storn, 1997). Simple GA uses a binary coding for representing problem parameters whereas DE uses real coding of floating point numbers. Among the DE's advantages are its simple structure, ease of use, speed and robustness. It can be used

Improvements on DE

When using any population based search algorithm in general and DE in particular to optimize a function, an acceptable trade-off between convergence rate (with reference to locating optimum) and robustness (with reference to not missing the global optima) must generally be determined. Convergence rate implies a fast convergence although it may be to a local optimum. On the other hand, robustness guarantees a high probability of obtaining the global optimum. A few attempts have already been made

Modified differential evolution (MDE)

The principle of modified DE is same as DE. The major difference between DE and MDE is that MDE maintains only one array (Fig. 1, Fig. 2). The array is updated as and when a better solution is found. Also, these newly found better solutions can take part in mutation and crossover operation in the current generation itself as opposed to DE (where another array is maintained and these better solutions take part in mutation and crossover operations in next generation). Updating the single array

Bench mark test functions

The reliability and efficiency of DE and MDE are tested and compared for several multimodal functions, which were used in earlier literature (Teh & Rangaiah, 2003). The selected benchmark functions are briefly described below and details of the global minimum are summarized in Table 1.

Optimization of selected non-linear chemical processes

The optimization of non-linear constraint problems is relevant to chemical engineering practice (Salcedo, 1992, Floudas, 1995). Non-linearities are introduced by process equipment design relations, by equilibrium relations and by combined heat and mass balances. There are many chemical processes which are highly non-linear and complex with reference to optimal operating conditions with many equality and inequality constraints. In this paper the following processes are considered for applying DE

Constraint handling

In Sections 5 Bench mark test functions, 6 Optimization of selected non-linear chemical processes, we have discussed the various test functions and selected chemical engineering problems. Most of the engineering optimization problems are constrained. The difficulty of using EAs in the constrained optimization is that the evolutionary operators used to manipulate the individuals of the population often produce solutions which are unfeasible. There are many methods to handle it. The handling of

Results and discussion

In Section 7, we have discussed the methods used for handling bound violation and constraints in the present study. The following subsections discuss the results obtained using DE and MDE for test functions followed by selected optimization problems from chemical engineering. Extensive computational comparisons have been made for all the chemical engineering problems considered using standard statistical hypothesis testing methods such as t-test.

Conclusions

The Modified Differential Evolution (MDE) algorithm has been introduced and compared to Differential Evolution (DE) for global optimization of benchmark test functions and selected non-linear chemical processes. Extensive computational comparisons have been made for all the chemical engineering problems considered using standard statistical hypothesis testing methods such as t-test. The results stated above clearly show the improvement upon the performance characteristics of DE with regard to

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Modified differential evolution (MDE) for optimization of non-linear chemical processes

Abstract

Introduction

Section snippets

Differential evolution (DE) in brief

Improvements on DE

Modified differential evolution (MDE)

Bench mark test functions

Optimization of selected non-linear chemical processes

Constraint handling

Results and discussion

Conclusions

Computers and Chemical Engineering

Computers and Chemical Engineering

Chemical Engineering Science

Computers and Chemical Engineering

Computers and Chemical Engineering

Computers and Chemical Engineering

Computer and Chemical Engineering

Computers and Chemical Engineering

Computers and Chemical Engineering

Evolutionary computation for global optimization of non-linear chemical engineering processes

Process plant simulation

Optimization of non-linear chemical processes using evolutionary algorithm

A differential evolution approach for global optimization of MINLP problems.

Selected applications of non-linear programming

Ant direction hybrid differential evolution for solving large capacitor placement problems

IEEE Transactions on Power Systems

Optimization for engineering design: Algorithms and examples.

Multi-objective optimization using evolutionary algorithms

A set of geometric programming test problems and their solutions

Mathematical Programming

Optimization of chemical processes

Non-linear and mixed-integer optimization