A new genetic algorithm based on prenatal genetic screening (PGS-GA) and its application in an automated process flowsheet synthesis problem for a membrane based carbon capture case-study

https://doi.org/10.1016/j.cherd.2017.10.009Get rights and content

Highlights

  • A new optimization algorithm called Prenatal Genetic Screening Genetic Algorithm.

  • Promising computational optimisation based on prenatal genetic screening (PGS).

  • Applied new algorithm (PGS-GA) to membrane-based carbon capture flowsheet design.

  • Showed that PGS-GA outperforms standard GA in lower number of generations.

  • PGS-GA led to 2.3% improvement in objective function value over GA algorithm.

Abstract

The genetic algorithm (GA) is a widely used optimization algorithm that mimics the process of natural selection to search and find useful solutions among sets of generations. In GA, generations are sets of possible solutions, known as chromosomes, which consist of a set of manipulating parameters, known as genes. In this study we introduce a new optimization algorithm called ‘Prenatal Genetic Screening’ Genetic Algorithm (PGS-GA) and investigate the comparative performance of this new algorithm against standard GA. This new evolutionary computation technique mimics the PGS procedure whereby a trained surrogate model is used to estimate the performance of each possible generated individual in GA generations and then diagnoses and replaces weak fetuses with stronger individuals. Firstly, the performance of this new algorithm is investigated on a set of known benchmark functions to assess the closeness to the global optimum in least number of generations as compared to GA. The results reveal that PGS-GA shows an efficient performance for optimization of multivariable multimodal functions and leads to noticeable improvements in convergence speed and closeness of the solution to the global optimum in all studied benchmark functions. This new optimization technique is then implemented for our automated process synthesis algorithm to generate the optimum process flowsheet for a membrane based CO2 capture process. It is shown that using PGS-GA leads to 2.3% improvement in the value of the objective function (product CO2 purity) over the GA algorithm. In addition, the presence of repeated flowsheets (structure, operating and design condition) among different solutions achieved using the algorithm starting from different randomly generated starting points that provide higher objective function values, approximately implies closeness of the solution to the global optimum. This consistency of the algorithm brings about a more robust flowsheeting algorithm that can provide higher performance solutions. Implementing more robust surrogate models, will facilitate the use of this algorithm in numerous other process design applications and beyond.

Introduction

The GA is based on evolutionary theory and is a commonly used optimisation method nowadays. This methodology is employed for optimization in a large variety of applications (Leong et al., 2016, Coelho Sampaio et al., 2016). GA is a stochastic (heuristic) optimization method or evolutionary computation process, in which a population of solutions is generated and it evolves over a sequence of generations based on biological principles of selection. This method mimics the process of natural selection to search and find useful solutions among sets of generations (Mayer, 1999). Directional selection is a type of natural selection that favours one extreme phenotype over the mean or other extreme. This procedure occurs during generations, and according to Darwinian theory causes the allele frequency to shift toward a special direction over time (Darwin and Bynum, 2009), where less fit chromosomes (solutions) are discarded, leaving behind only ‘stronger’ chromosomes. Directional selection modifies the genetic variation and changes allelic frequencies toward fitter populations eliminating or maintaining genetic variation. In GA, this concept is mainly the dominant selection algorithm where the fitness of each individual (solution) in each generation is evaluated, and solutions are selected for reproduction based on their fitness. Therefore, the fittest solutions in a natural process survive and are selected for reproduction while weak solutions are eliminated. In GA, genetic crossover operator is the algorithm or method through which offspring are generated from existing parents in order to maintain or improve the desired characteristics/genes, while rejecting undesired characteristics/genes. The significance of crossover mechanism is investigated and discussed in our previous work (Shafiee et al., 2016a). Genetic crossover is continued until convergence is achieved (Rios and Sahinidis, 2013).

GA has an acceptable performance and high efficiency for use with complex objective functions and for handling nonlinearities and constraints, but there is always a high risk of entrapment in local optima (Rios and Sahinidis, 2013, Ruiz et al., 2006, Li and Wang, 2007, Long et al., 2015). Therefore, there were some genetic efforts to improve the performance of GA by manipulating the structure of natural selection algorithm. Elitist GA is one of the early methods proposed to enhance GA population performance (Lin et al., 2000, Coello, 2000). Manipulating and changing crossover operator and offspring generation methods were also subjects of modifications based on fitness function in some studies (Srinivas and Patnaik, 1994, Shi et al., 2002, Laoufi et al., 2006). Xu et al. (2002) proposed a method called parent-offspring completion, in which parents are always put into competition with their offspring. Wu et al. proposed a modification on directional selection called ‘TDGA’ that removes 2/3 of the worst parent individuals and the best 1/3 are used. Then 1/3 of individuals are selected randomly from the removed parent individuals to be manipulated to reproduce offspring (Wu et al., 2015).

Surrogate model based optimization (SBAO) refers to the idea of using a surrogate model based on a set of known data to make an estimation for the objective function to speed up the optimization processes (Queipo et al., 2005, Jones et al., 1998, Tang et al., 2013, Queipo et al., 2002). This process is based on constructing an accurate surrogate model for the objective function based on a set of known data points, estimating the global optimum based on the optimization algorithm, evaluating the real value of objective function at estimated optima and investigation of convergence criteria (Queipo et al., 2005). Artificial neural network (ANN) is a powerful mathematical structure to make an estimation for the nonlinear relationships between input and output data sets of a complex function. ANN models showed an efficient performance in complex problems where identifying the characteristics of processes are difficult to describe using physical equations. This mathematical structure can establish an acceptable relationship between input and outputs in different systems and processes (Malviya and Pratihar, 2011). Therefore, ANN has been widely used for surrogate models to reduce the complexity of modelling of many structures and units (Henao and Maravelias, 2011, Sabuncuoglu and Touhami, 2002, Nascimento et al., 2000, Fahmi and Cremaschi, 2012, Adib et al., 2013). SBAO effectively increased the convergence speed of optimisation algorithms, but since the procedure of optimization is based on the estimation of surrogate model, the accuracy of this method depends on the accuracy of the surrogate model. An inappropriate surrogate model and objective function estimation, can misadjust the procedure of an optimization algorithm and put it at risk of convergence to local optima. Large diversity of training data is required to enable surrogate model to make an accurate estimation, and so it follows that, in complex problems, achieving this set of data might reduce the speed of the entire optimization procedure.

Prenatal genetic screening (PGS) is introduced in this paper for the first time as a computational analogue to real PGS. The term PGS refers to a set of diagnostic techniques for sample collection and analysis with the aim of detecting fetal anomalies in utero. Essentially, PGS tests provide an estimate of the likelihood of identifying pregnancies at risk or what the baby’s chances are of having an abnormality later in life. As an outcome, prenatal screening tests the ‘fitness’ of the fetus and offers parents the choice for termination of a pregnancy in case of serious diseased fetus (Rothenberg, 1994) that could lead to serious harm to fetus and mother, albeit the ethical considerations around this which are outside our computational discussion here.

In this current paper we use the concepts of PGS and SBAO to modify the directional selection mechanism in GA and apply this to the process flowsheeting problem. We used an ANN-based surrogate model as a diagnostic technique to investigate the structure of each chromosome (solution) before carrying out any evaluation. Thus enabling the algorithm to prevent production of weak individuals, and therefore ensuring a stronger generation is expected. Since, the produced elite individuals of each generation are expected to lead to more elite individuals in the next generation, the trend of evaluation is expected to shift toward the global optimum in a faster convergence.

This new self-directed ‘prenatal genetic screening’ genetic algorithm (PGS-GA) is used here to govern the optimization procedure toward the global optimum (Fig. 2). During this procedure, the first randomly generated individuals of the first generation in GA are evaluated based on a defined objective function. This set of data is used to train an ANN-based surrogate model to enable this model to be used to diagnose the possible performance of the following individuals. This surrogate model evaluates and screens undesired individuals based on estimated performances, and those eliminated individuals are replaced by new individuals that could meet the intended screening criteria. The accepted individuals accompanied by the first training set of data, generate the first generation in the GA. The surrogate model supervises and directs the crossover operator performance to produce high quality offspring. This surrogate model is modified and evaluated after generation of a set of new data based on new individuals and their corresponding objective function values. In PGS-GA, after generation and evaluation of a set of new data (accepted individuals and their corresponding objective function values), these data are added to the previous set of surrogate model data. Then, the surrogate model is updated (trained) again, based on the new set of data in each stage. It means that by progress of PGS-GA from one generation to the next one, and the increase of the number of training set of data, the surrogate model can cover a wider search space. This fact leads to robustness of the surrogate model and a better performance in estimation of objective function value.

Automated process flowsheet synthesis refers to systematic computational techniques for the design of process flowsheets that consider design and operation variables as well as equipment types (Barnicki and Siirola, 2004, Douglas, 1985, Nishida et al., 1981, Westerberg, 2004). These methods can be used to create optimised flowsheets based on optimal interconnections among process equipment (structure), optimal operating unit design (design) and, optimal unit operating conditions (parameters) (Yuan et al., 2013). Traditionally, such synthesis is performed by applying techniques based on engineering judgment and the process is then optimized using operating unit design and operating conditions. Such conventional techniques are developed over years through empirical process design and trial-and-error approaches. However, the achieved optimum design and parameters are only valid for the same process flowsheet, challenging their applicability over wider possibilities of flowsheet designs. In consequence, there is a considerable uncertainty that the achieved optimum point ends up being far from the global optimum (Shafiee et al., 2016a). Moreover, in today’s highly competitive markets, systematic computational synthesis and design of optimal process flowsheets is of critical significance to convert raw materials into desired products where energy efficiency and performance play key roles in lowering production costs (Sánchez and Cardona, 2012).

Another approach for flowsheet synthesis is to screen a wide range of possible flowsheet alternatives within a predefined superstructure (Grossmann et al., 2000, Grossmann et al., 1999) using optimization techniques such as mathematical programing (e.g. mixed integer linear or non-linear programming (Türkay and Grossmann, 1996, Trespalacios and Grossmann, 2014, Achenie and Biegler, 1990, Agarwal et al., 2010)) and evolutionary algorithms (e.g. differential evolution approach (Angira and Babu, 2006, Seader and Westerberg, 1977)). This is called optimization-based process synthesis. Since the first trials of optimization-based process synthesis in the late 60’s (Thompson and King, 1972, Hendry et al., 1973), many studies have been carried out to find the optimum process structure and design in various fields of chemical engineering including power systems (Zheng and Kim, 2013), heat exchanger networks (Cerda and Westerburg, 1983, Escobar and Trierweiler, 2013, Liu et al., 2014, Ciric and Floudas, 1989, Yee and Grossmann, 1991), water network (Hwang and Moore, 2011, Khor et al., 2012), separation processes with distillation columns (Floudas and Paules, 1988, Paules and Floudas, 1992, Gutiérrez-Guerra et al., 2014), refrigeration systems (Shelton and Grossmann, 1986a, Shelton and Grossmann, 1986b, Zhang and Xu, 2011, Wu and Zhu, 2002), evaporation systems (Hillenbrand, 1984) and crystallization processes (Lin et al., 2008) to name a few.

Generally, superstructure-based flowsheet synthesis algorithm is associated with two main challenges:

  • Postulating an inclusive superstructure.

  • Selecting and implementing an efficient optimization algorithm.

There are three methods available for superstructure generation, namely alternative collection, combinatorial synthesis, and insight-based synthesis (Quaglia et al., 2015). The alternative collection mechanism is based on collection and reorganization of all known and reported flowsheet variations in industrial and scientific literature by the designer to synthesise the most promising superstructure (Rojas-Torres et al., 2013). While this procedure relies heavily on existing knowledge, by applying this method there is a high chance that unknown high performance structures are neglected. In combinatorial synthesis mechanism, theoretically, the focus is on finding innovative designs by including all possible connections among material streams and operating units. Although holistic, this method leads to extremely large search spaces, especially for big flowsheets or in cases where more than two splits are considered for material streams and/or operation units design. This further leads to significant increase in computational complexity and costs. This method is the basis for Solution Structure Generation (SSG) method in P-graph method (Friedler et al., 1993, Friedler et al., 1995, Friedler et al., 1998) and mathematical programing based global optimization (Karuppiah and Grossmann, 2006, Tay et al., 2011). Insight-based synthesis mechanisms employ a synthesis algorithm to generate a technically feasible superstructure using two strategies: elimination of non-feasible or not-convenient alternatives from a relatively inclusive search space or inclusion of feasible connections among operating equipment or tasks. These strategies are based on engineering and commercial insights. The insight-based synthesis mechanism is the basis for many works such as Accelerated Branch and Bound (ABB) method (Friedler et al., 1996), decision mapping methods (Friedler et al., 1995), implicit superstructure generation (McCarthy et al., 1998) or works of Sirrola and Barnicki (Barnicki and Siirola, 2004, Siirola, 1996) and Floudas and Lin (Floudas and Lin, 2005, Floudas et al., 2012). The major disadvantage of this approach is the possibility of neglecting some alternative designs, which might lead to better outcomes, although with lower probability compared to the former heuristics-based method.

However, with the advent of ever-increasing computation power, automated process generation algorithms along with novel, and robust optimization methods, we expect that true and unbiased process flowsheet synthesis in an extensive search space are likely to become the norm.

Recently, in our previous work, a novel automated process flowsheet synthesis was proposed and developed (Shafiee et al., 2016a, Shafiee et al., 2016b, Shafiee et al., 2017). This flowsheeting approach is a systematic, computational, optimization-based and superstructure-free technique that considers the design and operation variables as well as equipment types with minimum human interference or bias. This methodology is an attempt to take advantage of a powerful computer-based computational algorithm to create an optimized flowsheet based on optimal interconnections among different equipment (structure), optimal operating unit design (design) and optimal unit operating condition (parameters). In this approach, instead of screening a set of known alternatives, defined as the superstructure, this algorithm is free to generate, evaluate and optimize through an undefined search space. In other words, the possible solutions in this method are randomly and freely generated through the entire search space, evaluated and modified through an optimization algorithm. The block diagram of the automated process flowsheet synthesis algorithm is illustrated in Fig. 1.

In this approach, instead of defining a completely known superstructure that depicts possible alternatives that are defined for the problem, it employs a set of preventive constraints. These constraints comprise physical, chemical, thermodynamic and any other engineering and scientific information required to determine the feasibility of a process flowsheet (see Section 2.4). These rules are defined as preventive criteria in generating procedures, in order to investigate the feasibility of structures. This approach is based on maximizing freedom in generation, therefore, as long as a generated alternative does not violate any of these rules (non-feasible flowsheet), such alternative flowsheet is among possible solutions in the search space of this algorithm.

The proposed automated process flowsheet synthesis algorithm is an optimization-based algorithm which takes advantage of a stochastic optimization algorithm to search a wide search space to select an optimal process flowsheet through an evolutionary procedure. Stochastic optimization algorithms are a group of non-traditional algorithms that attempt to reach the global optimum through successive solution selections and corrections. However, there is a wide variety of different optimization algorithms, and no special priority criterion exists in selection of an efficient optimization algorithm. The performance of each optimization algorithm, relies on problem characteristics such as objective function and case study (Shafiee et al., 2016a).

Many optimization algorithms such as, genetic algorithm (GA) (Holland, 1975), particle swarm optimization (PSO) (Zhan et al., 2009), ant colony optimization (ACO) (Dorigo et al., 1996), and others, were introduced and developed to find the optimum value within the search space in minimum time and with highest accuracy. GA employs a series of operators that are inspired by the natural evolution process to search and find useful solutions among sets of generations that contain possible solutions. In order to improve the quality of solutions through natural selection process, GA attempts to maintain desired genes and reject undesired ones through the generations. Since the performance of a flowsheet relies on presence of a series of structures or unit specifications, employing GA can be an advantageous tool to determine these structures in elite solutions of a generation and transfer them (desired genes) to the next generation. Then GA can propose an optimal flowsheet that contains many of these appropriate structures. Moreover, GA is usually independent of characteristic information and parameters of the problem or designing insight. Therefore, it can be promising to be employed for an un-biased strategy to select an optimal flowsheet with minimum human-based insight.

We used GA as the governing optimization algorithm for designing a high performance multistage membrane-based carbon capture plant. That methodology generated different high performance flowsheets that meet the constraints and produce high purity CO2 permeate. Furthermore, comparing different proposed solutions, the common features among them revealed some unpredicted hidden rules in process flowsheet synthesis for consideration to achieve a high purity permeate. These recurring hidden structures in the proposed flowsheets are translated to desired genes that are maintained through GA progress (Shafiee et al., 2016a).

This study is an attempt to serve the goal of achieving a more robust automated process flowsheet synthesis algorithm by manipulating the architecture of the governing optimization algorithm. In this context, the new optimization algorithm inspired by ‘prenatal genetic screening’ (PGS-GA) is designed and implemented as the governing optimization algorithm of the flowsheeting methodology. Therefore, we intend to reach high performance and similar flowsheets through a wide search space that depicts the global optimum flowsheet in the considered search space (Fig. 3).

In this study, we use and adopt the same case study we used in our previous work (viz. multistage membrane-based carbon capture plant — Fig. 4) and we apply PGS-GA to investigate the performance of this new algorithm as the governing optimization algorithm for the ‘synthesis’ of an optimal flowsheet rather than ‘screening’ a set of predefined flowsheets or superstructures (Shafiee et al., 2016a). The PGS-GA algorithm determines the best possible flowsheet based on a specified objective function and predefined constraints. In this approach, human bias and intervention in flowsheet generation is minimized. However, this algorithm employs a random generation algorithm to generate a wide variety of possible flowsheets, an ANN-based surrogate model is in charge of evaluation of the flowsheets before the rigorous simulation step. This model diagnoses the performance of the flowsheets based on their previous performance and conducts a PGS step to eliminate weak flowsheets (Fig. 2). This procedure enhances the performance of the new generation and it is expected that, using this approach, to observe a high performance of PGS-GA in achieving global optimum. However, finding global optimum is practically impossible, a solution acceptance rule based on solution repeatability and maximum value of objective function is used to determine if the achieved solution is near the global optimum. It will be evident that the proposed method with required modifications is applicable for any other problem.

Section snippets

Methodology

Genetic algorithm (GA) is one of the commonly used optimisation methods that has an acceptable performance and high efficiency in complex objective functions in terms of nonlinearities and constraints imposed. In this study we outlined a self-directed Prenatal Genetic Screening Genetic Algorithm (PGS-GA) to govern the optimization procedure toward global optimum (Fig. 2). This procedure is based on training a surrogate model based on generated and evaluated individuals and using the high

The performance of the PGS-GA algorithm using benchmarking functions

Before using any new proposed algorithm for a desired application, usually the performance of this algorithm is investigated using a series of standard and known benchmark functions (Rada-Vilela et al., 2014). Therefore, in this study the performance of this algorithm is investigated using a set of benchmark functions. The results of PGS-GA optimization algorithm for these functions are subject of comparison with GA results. The capability of the new proposed algorithm in exploring the global

Application of PGS-GA on the flowsheet synthesis problem

The main purpose of any optimization algorithm is reaching or approaching sufficiently to the global optimum. Additionally, there are some other requirements that should be met in every accepted and effective approach. It is expected to see a systematic and stable improvement of the fitness function of the best individuals through the progress of the optimization algorithm generations. Moreover, beside accuracy, speed of convergence is another important factor for each optimization algorithm.

Conclusion

Previously, we introduced an optimization-based (GA) process flowsheet synthesis algorithm and employed it for two example applications. This current study is an attempt to increase the robustness of that algorithm by manipulating and improving the architecture of the governing optimization algorithm.

In this study, a new optimization algorithm called Prenatal Genetic Screening Genetic Algorithm (PGS-GA) was introduced. This new optimization algorithm was investigated through a comparative study

References (107)

  • S.-C. Horng et al.

    Evolutionary algorithm assisted by surrogate model in the framework of ordinal optimization and optimal computing budget allocation

    Inf. Sci.

    (2013)
  • D. Karaboga et al.

    A comparative study of artificial bee colony algorithm

    Appl. Math. Comput.

    (2009)
  • R. Karuppiah et al.

    Global optimization for the synthesis of integrated water systems in chemical processes

    Comput. Chem. Eng.

    (2006)
  • C.S. Khor et al.

    A superstructure optimization approach for water network synthesis with membrane separation-based regenerators

    Comput. Chem. Eng.

    (2012)
  • C.C. Leong et al.

    Genetic algorithm optimised chemical reactors network: a novel technique for alternative fuels emission prediction

    Swarm Evol. Comput.

    (2016)
  • S.W. Lin et al.

    Synthesis of crystallization processes for systems involving solid solutions

    Comput. Chem. Eng.

    (2008)
  • X.-w. Liu et al.

    Studies on the retrofit of heat exchanger network based on the hybrid genetic algorithm

    Appl. Therm. Eng.

    (2014)
  • Q. Long et al.

    A genetic algorithm for unconstrained multi-objective optimization

    Swarm Evol. Comput.

    (2015)
  • M. Mayer

    A network parallel genetic algorithm for the one machine sequencing problem

    Comput. Math. Appl.

    (1999)
  • E. McCarthy et al.

    An automated procedure for multicomponent product separation synthesis

    Comput. Chem. Eng.

    (1998)
  • C.A.O. Nascimento et al.

    Neural network based approach for optimization of industrial chemical processes

    Comput. Chem. Eng.

    (2000)
  • H. Nazif et al.

    Optimised crossover genetic algorithm for capacitated vehicle routing problem

    Appl. Math. Modell.

    (2012)
  • V. Ojalehto et al.

    Agent assisted interactive algorithm for computationally demanding multiobjective optimization problems

    Comput. Chem. Eng.

    (2015)
  • B. Ostadmohammadi Arani et al.

    An improved PSO algorithm with a territorial diversity-preserving scheme and enhanced exploration–exploitation balance

    Swarm Evol. Comput.

    (2013)
  • G.E. Paules et al.

    Stochastic programming in process synthesis: a two-stage model with MINLP recourse for multiperiod heat-integrated distillation sequences

    Comput. Chem. Eng.

    (1992)
  • A. Quaglia et al.

    Systematic network synthesis and design: problem formulation, superstructure generation, data management and solution

    Comput. Chem. Eng.

    (2015)
  • N.V. Queipo et al.

    Surrogate modeling-based optimization for the integration of static and dynamic data into a reservoir description

    J. Petrol. Sci. Eng.

    (2002)
  • N.V. Queipo et al.

    Surrogate-based analysis and optimization

    Prog. Aerosp. Sci.

    (2005)
  • J. Rada-Vilela et al.

    Population statistics for particle swarm optimization: Resampling methods in noisy optimization problems

    Swarm Evol. Comput.

    (2014)
  • R. Ruiz et al.

    Two new robust genetic algorithms for the flowshop scheduling problem

    Omega

    (2006)
  • Ó.J. Sánchez et al.

    Conceptual design of cost-effective and environmentally-friendly configurations for fuel ethanol production from sugarcane by knowledge-based process synthesis

    Bioresour. Technol.

    (2012)
  • A. Shafiee et al.

    Automated process synthesis for optimal flowsheet design of a hybrid membrane cryogenic carbon capture process

    J. Clean. Prod.

    (2017)
  • M.R. Shelton et al.

    Optimal synthesis of integrated refrigeration systems—II: implicit enumeration scheme

    Comput. Chem. Eng.

    (1986)
  • J.J. Siirola

    Strategic process synthesis: advances in the hierarchical approach

    Comput. Chem. Eng.

    (1996)
  • A.W. Westerberg

    A retrospective on design and process synthesis

    Comput. Chem. Eng.

    (2004)
  • L. Achenie et al.

    A superstructure based approach to chemical reactor network synthesis

    Comput. Chem. Eng.

    (1990)
  • D.H. Ackley

    An empirical study of bit vector function optimization

    Genet. Algorithms Simul. Annealing

    (1987)
  • A. Agarwal et al.

    A superstructure-based optimal synthesis of PSA cycles for post‐combustion CO2 capture

    AIChE J.

    (2010)
  • T. Bäck et al.

    An overview of evolutionary algorithms for parameter optimization

    Evol. Comput.

    (1993)
  • S.D. Barnicki et al.

    Systematic chemical process synthesis

    Formal Engineering Design Synthesis

    (2001)
  • S.D. Barnicki et al.

    Process synthesis prospective

    Comput. Chem. Eng.

    (2004)
  • M. Bhattacharya

    Meta model based EA for complex optimization

    (2008)
  • I.O. Bohachevsky et al.

    Generalized simulated annealing for function optimization

    Technometrics

    (1986)
  • C.A. Coello

    An updated survey of GA-based multiobjective optimization techniques

    ACM Comput. Surv.

    (2000)
  • C. Darwin et al.

    The origin of species by means of natural selection: or, the preservation of favored races in the struggle for life

    (2009)
  • L.D. Davis
    (1991)
  • K.A. De Jong

    Analysis of the behavior of a class of genetic adaptive systems

    (1975)
  • M. Dorigo et al.

    Ant system: optimization by a colony of cooperating agents

    IEEE Trans. Syst. Man Cybern. B: Cybern.

    (1996)
  • J. Douglas

    A hierarchical decision procedure for process synthesis

    AIChE J.

    (1985)
  • Á.E. Eiben et al.

    Performance of Multi-parent Crossover Operators on Numerical Function Optimization Problems

    (2017)
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