Cuckoo Search Optimization and Application to LiquidLiquid Equilibrium
Introduction
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This chapter uses a new variant of the optimization technique, namely Cuckoo Search (CS) algorithm, to generate the liquidliquid equilibrium (LLE) data. Liquidliquid extraction is an important separation technology with a wide range of applications in chemical, petrochemical and pharmaceutical industries. The LLE data of multicomponent systems are essential for proper understanding of the extraction process and for the designing and optimization of separation processes. Excess Gibbs free energy models, such as the nonrandom twoliquid (NRTL; Renon & Prausnitz, 1968) and the UNIversal QUAsiChemical (UNIQUAC; Abrams & Prausnitz, 1975) models, are commonly used to predict the LLE as they provide good agreement with experimental data (Banerjee, Singh, Sahoo, & Khanna, 2005; Santiago, Santos, & Aznar, 2009; Vatani, Asghari, & VakiliNezhaad, 2012). For LLE prediction, each of these models requires binary interaction parameters. These parameters are generally estimated from the known experimental LLE data by the optimization of an objective function. Mathematically, the aim of optimization is to find the set of inputs that either maximizes or minimizes the output of the objective function. In LLE modelling, the objective function is nonlinear and highly nonconvex having multiple local optima which makes most conventional methods (deterministic algorithms) inefficient and stuck in the wrong solutions. For the correct LLE prediction in liquidliquid phase equilibria, finding the global optimum (reliable interaction parameters) thus becomes a necessary requirement.
Heuristic and metaheuristic algorithms are designed to deal with highly nonlinear and nonconvex problems. Most of these algorithms are natureinspired or bioinspired. Natureinspired metaheuristic algorithms are becoming increasingly popular to solve global optimization problems. All stochastic algorithms with randomization and local search can be termed as metaheuristic algorithms. The two major components of any metaheuristic algorithm are: selection of the best solutions which ensures that the solutions will converge to the optimal and randomization which avoids the solutions being trapped at the local optima. Their efficiency is remarkable; these have many advantages over traditional, deterministic methods and thus have been applied in almost all areas of science, engineering and industry (Yang & Deb, 2014). Recent literature report many established nature inspired metaheuristics, which are listed in Table 7.1. These algorithms are broadly classified into evolutionary algorithms, physical algorithms, swarm intelligence, bioinspired algorithms and others (Nanda & Panda, 2014).
Genetic algorithm (GA), developed by John Holland and his collaborators, is based on Charles Darwin's theory of natural selection and remains one of the most widely used optimization algorithms in modern nonlinear optimization. Many variants of the GA have been developed and applied to a wide range of optimization problems. The GA starts with a set of randomly generated solutions called population. Then, the fitness of all the individuals in the population is evaluated and a new population is created by performing selection, crossover and mutation operations. The algorithm continues to evolve the population till certain stopping criteria are met (Yang, 2010).
Simulated annealing is a trajectorybased search algorithm which mimics the annealing process in material processing. Annealing is a process in metallurgy where metals are heated to specific high temperature and then slowly cooled to decrease defects and make them reach a state of low energy where they are very strong. After each iteration of the simulated annealing algorithm, a new solution is randomly generated. The algorithm accepts not only
TABLE 7.1
Broad Classification of Nature Inspired Metaheuristic Algorithms
Types 

Evolutionary algorithms 
Genetic algorithm (GA) Differential evolution (DE) Genetic programming (GP) 
Physical algorithms 
Simulated annealing (SA) Harmony search (HS) 
Swarm intelligence 
Particle swarm optimization (PSO) Ant colony optimization (ACO) Artificial bee colony (ABC) 
Bioinspired algorithms 
Artificial immune system (AIS) Krill herd algorithm 
Other natureinspired algorithms 
Firefly algorithm Cuckoo Search algorithm Bat algorithm 
all new solutions that lower the objective function but also, with a certain probability, the solutions that raise the objective function. By accepting solutions that raise the objective, the algorithm avoids being trapped in local minima in early iterations and is able to explore globally for better solutions (Yang, 2010). Particle swarm optimization (PSO) is inspired by the swarm intelligence of fish and birds which has been discussed in detailed in Chapter 6.
The application of natureinspired metaheuristic algorithms for solving phase equilibrium calculations (PEC), phase stability (PS) problems and for parameter estimation (PE) has grown considerably in recent years (Bonilla Petriciolet, Rangaiah, & SegoviaHernandez, 2010; BonillaPetriciolet & SegoviaHernandez, 2010; Fateen, BonillaPetriciolet, & Rangaiah, 2012; FernandezVargas, BonillaPetriciolet, & SegoviaHernandez, 2013; Srinivas & Rangaiah, 2007; Zhang, FernandezVargas, Rangaiah, BonillaPetriciolet, & SegoviaHernandez, 2011; Zhang, Rangaiah, & BonillaPetriciolet, 2011). Apart from the PEC and PS problems, stochastic global optimization methods have also been applied for the estimation of binary interaction parameters in multicomponent LLE systems. Singh, Banerjee and Khanna (2005) utilized the GA to estimate the binary interaction parameters for the NRTL and the UNIQUAC models in multicomponent LLE systems and demonstrated that their performance was better than the inside variance estimation method and the techniques applied in ASPEN and DECHEMA. Sahoo, Banerjee, Ahmad and Khanna (2006) calculated the interaction parameters for the NRTL model in ternary, quaternary and quinary LLE systems based on the GA and showed that the results obtained using the GA were better than other techniques in the literature. Ferrari, Nagatani, Corazza, Oliveira, and Corazza (2009) applied SA and PSO algorithms for PE of the NRTL and the UNIQUAC models for binary and multicomponent LLE systems and showed that both algorithms were capable of modelling liquidliquid equilibrium data. Merzougui, Hasseine, Kabouche, and Korichi (2011) used a hybrid algorithm, that is, a combination of the GA and the LevenbergMarquardt (LM) method, for PE with the NRTL and the UNIQUAC models for six LLE systems and found good correlation with experimental data. Vatani et al.
 (2012) also performed the LLE calculation for 20 different ionic liquid (IL) based ternary LLE systems by the NRTL and the twosuffix Margules models with binary interaction parameters calculated using the GA. A harmony search (HS) algorithm was used by Merzougui, Hasseine and Laiadi (2012) to calculate the interaction parameters of the NRTL model for 20 ternary liquid liquid systems. Kabouche, Boultif, Abidi and Gherraf (2012) have used SA, GA, NelderMead Simplex (NMS), SANMS (hybrid) and GANMS (hybrid) to estimate interactions parameters of the NRTL and the UNIQUAC models for LLE systems. The hybrid algorithm GANMS showed the best performance in terms of rootmeansquare deviation (RMSD) with minimum number of iterations among the others. BonillaPetriciolet, Fateen and Rangaiah
 (2013) have recently analyzed the capabilities of seven stochastic global optimization methods, SA, GA, DE, PSO, HS, differential evolution with tabu list (DETL) and bare bones PSO (BBPSO), to model mean activity coefficients in aqueous solutions of quaternary ammonium salts at 25°C using the electrolyte NRTL model. The results indicated that SA, DETL and BBPSO offer better performance for solving PE problems involved in the modelling of the thermodynamic properties of ILs.
Besides the wellknown methods, the investigations on natureinspired optimization algorithms are still currently under development. CS is one of the latest natureinspired metaheuristic algorithms developed by Yang and Deb (2009). It is a populationbased method which mimics the breeding behavior such as brood parasitism of certain cuckoo species. This algorithm is enhanced by the socalled Levy flights. Recent studies have shown that the CS algorithm is potentially far more efficient than other algorithms in many applications. It has been applied in many areas (Fister, Fister, & Fister, 2013) which include applied thermodynamic calculations. Bhargava, Fateen and BonillaPetriciolet (2013) have applied CS algorithm for solving PS, phase equilibrium and reactive phase equilibrium problems. They found that CS offers a reliable performance for solving these thermodynamic calculations and is better than other metaheuristics for phase equilibrium modelling. Fateen and BonillaPetriciolet (2014a) have compared the reliability and efficiency of eight promising natureinspired metaheuristic algorithms for the solution of nine difficult PS and phase equilibrium problems. These algorithms are the Cuckoo Search (CS), intelligent firefly (IFA), bat (BA), artificial bee colony (ABC), MAKHA (a hybrid between monkey algorithm and krill herd algorithm), covariance matrix adaptation evolution strategy (CMAES), magnetic charged system search (MCSS) and BBPSO. The results clearly showed that CS is the most reliable of all tested optimization methods as it successfully solved all thermodynamic problems tested in the study. Fateen and BonillaPetriciolet (2014b) have also applied gradientbased cuckoo search (GBCS) algorithm for solving several challenging PS problems and analyzed its performance at different numerical effort levels. The GBCS was found to perform better than the original CS algorithm. In comparison with other stochastic optimization, GBCS proved to be the most reliable without any reduction in efficiency.
The computation using the CS algorithm is scarce for multicomponent phase equilibria problems. In a recent work by JaimeLeal, BonillaPetriciolet, Bhargava and Fateen (2015), the CS algorithm was used to predict the binary phase equilibrium data for aqueous quaternary ammonium IL mixtures. In their work, the mean activity coefficients of quaternary ammonium ILs were predicted using the eNRTL model. The results obtained were very encouraging with a global success rate (SR) of ~86% with CS as compared to ~77% with other stochastic methods. However, the methods of computation with ternary, quaternary or quinary systems are not available in the literature. Keeping this limitation in mind, we have attempted to predict the multicomponent LLE data for both IL and organic solvent systems. In our study, binary interaction parameters of the UNIQUAC and the NRTL models were estimated using the CS algorithm for 39 ternary systems which include 32 ILbased systems and 7 organicsolventbased systems. The results (RMSD) thus obtained were compared with those reported in the literature. The performance of CS was compared with the GA and the PSO algorithms using three quaternary systems and one quinary system.
 [1] Sections 7.1 and 7.2 reprinted (adapted) with permission from A. Bharti, Prerna, T. Banerjee,Applicability of Cuckoo Search algorithm for the prediction of multicomponent liquidliquidequilibria for imidazolium and phosphoniumbased ionic liquids, Ind. Eng. Chem. Res. 54,1239312407, 2015. Copyright 2015 American Chemical Society.