An oracle penalty and modified augmented Lagrangian methods with firefly algorithm for constrained optimization problems

Abstract

Almost all engineering optimization problems in the real world are constrained in nature. Swarm intelligence is a bio-inspired technique based on studying and observing fireflies, ants, birds and fish in nature. Firefly algorithm (FA) is the most prominent swarm based metaheuristic algorithm used for solving a global optimization problem. This paper presents two new constrained optimization algorithms: (1) firefly algorithm with extended oracle penalty method (FA-EOPM) and (2) modified augmented Lagrangian with firefly algorithm (MAL-FA). These proposed algorithms are applied for solving classic thirteen benchmark constraint problems as well as a few good engineering problem designs. The efficiency, effectiveness, and performance of MAL-FA and FA-EOPM algorithms are estimated on the basis of statistical analysis such as best optimal value, worst value, mean value, p value and standard deviation value against the existing methods. The experimental results show that the proposed MAL-FA algorithm offers better outcomes for most of the cases in terms of the number of function evaluations compared to various optimization algorithms.

1 Introduction

In recent few years, non-traditional optimization methods, also called as evolutionary algorithms (EAs) have become the most powerful and efficient methodologies for solving complex optimization problems in various fields, including engineering (Tayarani et al. 2015), finance (Shao et al. 2014), bioinformatics (Sarkar and Maulik 2015), physics (Can and Alatas 2015) and manufacturing industry (Wari and Zhu 2016), primarily due to the composite nature of objective functions, nonlinear and problem-specific constraints. These methods include genetic algorithm (GA) (Goldberg and Holland 1988), differential evolution (DE) (Storn and Price 1997; Yuan-Long et al. 2015), evolution strategy (ES) (Beyer and Schwefel 2002), simulated annealing (SA) (Bandyopadhyay et al. 2008; Troyer 2016), particle swarm optimization (PSO) (Kennedy 2011), ant colony optimization (ACO) (Dorigo and Stützle 2010), firefly algorithm (FA) (Yang 2010), teaching learning-based optimization (TLBO) (Rao et al. 2011), biogeography-based optimization (BBO) (Simon 2008) etc. These population-based metaheuristic algorithms are more powerful than conventional based optimization algorithms. One of the most attractive field in population-based metaheuristic algorithm is swarm intelligence (SI) (Beni 2005). Recently, some of most popular swarm-based algorithms are: Moth search algorithm (Wang 2016), grey wolf optimizer (Mirjalili et al. 2014), krill herd algorithm (Gandomi and Alavi 2012; Wang et al. 2014a), stud krill herd algorithm (Wang et al. 2014b), chaotic krill herd algorithm (Wang et al. 2014d), A multi-stage krill herd algorithm (Wang et al. 2016c), Hybrid krill herd algorithm with DE (Wang et al. 2014c), the ant lion optimizer (Mirjalili 2015), monarch butterfly optimization (Wang et al. 2015c), elephant herding optimization (Wang et al. 2015b), the whale optimization algorithm (Mirjalili and Lewis 2016), earthworm optimization algorithm (Wang et al. 2015a), dolphin echolocation algorithm (Kaveh and Farhoudi 2013), chaotic cuckoo search (Wang et al. 2016a). Hybridization of metaheuristic algorithm, also called the hybrid-metaheuristic (Talbi 2002), specifically local search methods combine with population-based methods to get more desirable effective systems that utilize and combine advantages of the individual pure strategies. Some of hybrid metaheuristic algorithms are: harmony search algorithm with cuckoo search (Wang et al. 2016e), krill herd algorithm with cuckoo search (Wang et al. 2016d), krill herd algorithm with quantum-behaved particle swarm optimization (Wang et al. 2016b) Without loss of generality, non-linear constrained problems are defined as:

$${\text{Minimize}}\;\;f(X) = (x_{1} ,x_{2} , \ldots ,x_{n} ),\;\;X \in s,$$

(1)

$${\text{Subject to }}\quad g_{j} (X) \le 0\quad j = 1, \, 2, \, 3, \ldots ,p;$$

(2)

$$h_{j} (X) = 0\quad j = p + 1, \ldots ,q ;$$

(3)

$$l_{k} \le k \le u_{k}$$

(4)

where $(X) = (x_{1,} x_{2} , \ldots ,x_{n} )$ denotes the n-dimensional decision vector, $x_{k}$ is kth dimension (variable); $f(X)$ is the objective function as in (1), minimization function $l_{k}$ and $u_{k}$ are the lower and upper bounds values respectively, the search space defined as $s = \prod\nolimits_{k = 1}^{n} {[l_{k} } ,u_{k} ]$; the number of inequality and equality constraints are $p$ and $q - p$ respectively, $g_{j} (X)$ inequality constraints and $h_{j} (X)$ equality constraints.

The feasible solution that satisfies all linear and non-linear constraints. The constraints that satisfy $G_{j} (X^{*} ) = 0$ at the optimal value $X*$ are said to be active constraints. All linear constraints are active constraints. The linear constraints change into non-linear constraints by adding some other non-linear constraints as

$$G_{j} (X) = \left\{ {\begin{array}{*{20}l} {max\{ g_{j} (X),0\} } \hfill & {j = 1, \ldots ,p} \hfill \\ {\hbox{max} \{ |h_{j} (X)| - A,0\} } \hfill & {j = p + 1, \ldots ,q} \hfill \\ \end{array} } \right.$$

(5)

where ‘A’ is a positive value for linear constraint and $G_{j} (X) = \sum\nolimits_{j = 1}^{q} {G_{j} (X)}$ serves as the degree constraint violation of $X$ on all constraints.

This paper is well ordered as follows: Sect. 2 contain classification of constrained optimization algorithms (COPs); Sect. 3 deals with the basics of FA with penalty approaches; Sect. 4 presents step by step explanation of the proposed MAL-FA and EOPM-FA algorithms. Section 5 contains the computational results of the proposed algorithms with thirteen benchmark functions. Here the functions are described and compared with existing methods. The implementation of the proposed algorithms are examined with an engineering problem in Sect. 6. And eventually, the conclusions drawn are discussed in Sect. 7.

2 Classification of constrained optimization algorithms (COAs)

Normally, constrained optimization algorithms are of two types: (1) constraint handling techniques and (2) search algorithms such as GA, PSO, DE etc. To solve constraint optimization problems (COPs) in the field of bio-inspired (evolutionary) computation, various metaheuristic algorithms are developed. Due to the presence of constraint, several constraint handling approaches (CHA) are proposed to be used with EAs (Coello 2002; Mezura-Montes and Coello 2011; Michalewicz and Schoenauer 1996), while the search techniques employed is discussed as a separate issue. The primary aim of CHA is to calculate a criterion to compare the individuals in parent and offspring populations and COPs have roughly explored in five types of constraint handling approaches which exist within EAs (Mezura-Montes and Coello 2011):

• 1. Penalty functions approach (Albasri et al. 2015; Coit et al. 1996; Farmani and Wright 2003; Hamida and Schoenauer 2002) solves the constrained problems into an unconstrained problems by adding an infeasible penalty parameter to those solutions. These methods includes static penalty (Homaifar et al. 1994; Smith and Coit 1997), dynamic penalty (da Silva et al. 2010), adaptive penalty (Coit et al. 1996; Lemonge and Barbosa 2004; Tessema and Yen 2009), exact penalty (Di Pillo and Grippo 1989; Liu et al. 2016), oracle penalty (Schlüter and Gerdts 2010), augmented Lagrangian method (Birgin and Martínez 2014; Costa et al. 2012)and death penalty (Kramer 2010; Kramer and Schwefel 2006);

• 2. Preference of feasible solution over infeasible solution approach uses search operator for handling feasible and infeasible solution (Deb 2000; Mezura-Montes and Coello 2005; Takahama and Sakai 2005).

• 3. In hybrid methods (Talbi 2002), EAs are combined with heuristic rules or classical constrained search methods. This approach includes (1) combination of two methods like Otsu and Kapur with bacterial foraging and harmony search algorithm (Dehshibi et al. 2016); (2) simulated binary crossover and DE with polynomial mutation are combined (Lin et al. 2016); (3) PSO combined with GA (Garg 2016) and (4) hybrid firefly algorithm (HFA) (Zhang et al. 2016) combines FA and DE advantages and they can be executed in parallel to promote information sharing among the offspring and thus enhancing searching efficiency.

• 4. Decoder approaches (Jacquin et al. 2016; Koziel and Michalewicz 1999; Prestwich et al. 2015) uses mapping of feasible solution search space onto sample space where nature inspired algorithms (NIAs) can get better outcomes;

• 5. Multi-objective (vector) optimization approaches (Deb 2000; Jiang and Yang 2016a, b; Mezura-Montes et al. 2008; Silva et al. 2014) transforms a constraint optimization problem into vector problems with two or more objectives.

This paper contains hybridization techniques named as MAL-FA and FA-EOPM. These algorithms have been proposed by incorporating extended oracle penalty method and modified Augmented Lagrangian approach into FA to evaluate the nonlinear problems. The proposed algorithms MAL-FA and FA-EOPM are examined on well-known mathematical benchmark functions and engineering design optimization problems and the performance of the algorithm is compared with present existing algorithms in EAs.

3 FA with penalty approaches

This section is concerned about FA with two penalty approaches such as modified augmented Lagrangian method (MALM) and oracle penalty approach (OPM).

In literature, the performance of FA are tested using standard benchmark function for constraint problems (Łukasik and Żak 2009). Implementation of dynamic penalty approaches with the FA for solving non-linear global optimization problems are given in Francisco et al. (2015). An adaptive FA approach is proposed to solve mechanical design optimization problems considered in Baykasoğlu and Ozsoydan (2015). Use of feasibility-based rules with FA in sequence with the search of the optimal solution for constrained problems are addressed in Brajevic et al. 2012. Two new penalty-based approaches such as hyperbolic tangent function and the other inverse hyperbolic sine function are proposed by Costa et al. (2016) to solve benchmark functions using FA for global optimization.

3.1 Basic firefly algorithm (FA)

In 2007, Yang proposed a nature inspired FA (Yang 2010) and a detailed review of FA provided by Fister et al. (2013) in which brightness function is used as objective or fitness function. The flowchart of basic FA is shown in Fig. 1.

Fig. 1

Flow chart of basic FA

Using Euclidean a distance formulae, distance between any two fireflies i.e. ‘a’ and ‘b’ at $x_{a}$ and $x_{b}$ at Cartesian distance is stated by:

$$r_{ab} = |x_{a} - x_{b} | = \sqrt {\left( {\sum\nolimits_{k = 1}^{d} {(x_{a,k} - \quad x_{b,k} )^{2} } } \right)}$$

(6)

where $x_{a,k}$ = $k{th}$ component of spatial coordinate of $x_{a}$ of $a{th}$ firefly and $x_{b,k}$ = $k{th}$ component of spatial coordinate of $x_{b}$ of $b{th}$ firefly.

In 2D case:

$$r_{ab} = \left( {x_{a} - x_{b} } \right)^{2} + \left( {y_{a} - y_{b} } \right)^{2}$$

(7)

Typically, movement of the less attractive firefly “$a$” is gets attracted to different more attractive (brighter) firefly.

“$b$” is shown as:

$$x_{a} = x_{a} + \beta_{0} e^{{ - \gamma r^{2}_{ab} }} \left( {x_{b} - x_{a} } \right) + \alpha \left( {rand - 0.5} \right)$$

(8)

where 2nd term is to calculated the attractiveness of two fireflies, 3rd term is a random parameter, where ‘Rand’ is a used to generate random numbers uniformly between 0 and 1. In several cases, attractiveness parameter $\beta_{0}$ = 1 and randomness parameter α is in between 0 and 1.

3.2 Modified augmented Lagrangian approach

Augmented Lagrangian approach (Birgin et al. 2010; Birgin and Martínez 2014; Wright and Nocedal 1999) is used to evaluate the non-linear optimization problems.

If the minimum and maximum permissible value bounds (4) are not contained, it is can possible to use the modified augmented Lagrangian method (MALM) to evaluate problems (1)–(3). The unconstrained optimization subproblem at the kth step of this method is given in Long et al. (2016) and Shariff and Dormand (2003):

$${\text{Minimize}}\;\;G(X,\mu^{k} ,\rho^{k} )$$

(9)

where $G(X,\mu ,\rho )$ is the following MAL function:

$$G(X,\mu ,\rho ) = f(X) - \sum\limits_{j = p + 1}^{q} {\left[ {\mu_{j} h_{j} (X) - \frac{{\rho_{j} \left( {h_{j} (X)} \right)^{2} }}{2}} \right] - } \sum\limits_{j = 1}^{p} {\tilde{p}_{j} } (X,\mu ,\rho )$$

(10)

and $\tilde{P}(X,\mu ,\rho )$ is determined as:

$$\tilde{P}(X,\mu ,\rho ) = \left\{ {\begin{array}{*{20}l} {\mu_{j} g_{j} (X) - \frac{{\rho_{j} (g_{j} (X)^{2} )}}{2}} \hfill & {{\text{if }}\mu_{j} - \rho_{j} g_{j} (X) > 0} \hfill \\ {\frac{{\mu_{j}^{2} }}{{2\rho_{j} }}} \hfill & {} \hfill \\ \end{array} } \right.$$

(11)

such that $f(X)$ is an objective function, 2nd term is equality function and 3rd term is inequality function of MAL, $\mu$ are Lagrangian multipliers and $\rho$ is penalty parameter.

If the minimum and maximum permissible value bounds (4) are present, then one can use MALM to modify and minimize function given as in Liang et al. (2001):

$${\text{Minimize}}\;\;G(x,\mu^{k} ,\rho^{k} )\;\;{\text{such that}}\;\;l \le k \le u$$

(12)

where as $G(X,\mu ,\rho )$ is MAL function with lower and upper bounds.

3.3 Extended oracle penalty approach

In an oracle penalty approach (Schluter and Gerdts 2010) key ideas are provided for the transformation of objective function $f(X)$ of the problem (1)–(4) into equality constraint where $h_{0} (X) = f(X) - \varOmega = 0$, and the parameter omega ($\varOmega$) denotes an oracle coefficient. An objective function is unessential in the transformed problem definition and can be declared as a constant zero function $\tilde{f}(X)$. The transformed problem defines as:

$$\begin{aligned} & {\text{Minimize}}\quad \tilde{f}(X) = 0 \\ & {\text{subject}}\;{\text{to}}\;h_{0} (X) = f(X) - \varOmega = 0{\kern 1pt} ,\quad \varOmega \in {\mathbb{R}}, \\ & h_{j} (X) = 0,\quad j \, = 1,2,3, \ldots , \, p \in {\mathbb{N}}, \\ & g_{j} (X) \le 0,\quad j = p + 1, \ldots ,q \in {\mathbb{N}}, \\ \end{aligned}$$

(13)

If X* is a global optimal solution of problem (1), then oracle parameter $\varOmega$ = $f(X*)$ denotes the feasible solution of problem (13). Let suppose for a given constraint optimization problem the global optimal solution of objective function value $f(X*)$ is well known, the problem definition (13) holds a remarkable advantage compared to constraint problem (1). By transforming the objective function into an equality constraint, in which minimizing the new constraint $h_{0} (X)$ and minimizing the residual of the original constraint $h_{1} (X), \ldots ,h_{p} (X)$ becomes directly comparable. This comparability can be exploited by a penalty function, which balances its penalty weight on either the transformed objective function or the original constraints.

The basic oracle penalty function as given in

$$P(X) = \alpha \cdot \left| {f(X) - \varOmega } \right| + \left( {1 - \alpha } \right) \cdot res(X)$$

(14)

$$where{\kern 1pt} \,\alpha = \left\{ {\begin{array}{*{20}c} {1 - \frac{1}{{\sqrt {\frac{{\left| {f(X) - \varOmega } \right|}}{res(X)}} }},} & {if\;res(X) \le \left| {f(X) - \varOmega } \right|} \\ {\frac{1}{2}\sqrt {\frac{{\left| {f(X) - \varOmega } \right|}}{res(X)}} ,} & {if\;res(X) > \left| {f(X) - \varOmega } \right|} \\ \end{array} } \right.$$

(15)

where $f(X)$ is an objective function, $res(X)$ is a residue function which measures the constraint violations of constraint problems.

The extended oracle penalty approach $EOP(X)$ (Dong et al. 2014) is given as

$$EOP(X) = \left\{ {\begin{array}{*{20}l} {\alpha .f(X)\left| { - \varOmega } \right| + \left( {1 - \alpha } \right) \cdot res(X)} \hfill & {{\text{if}}\;f(X) \, > \varOmega {\kern 1pt} {\kern 1pt} \;{\text{or }}\;res(X) > 0} \hfill \\ { - \left| {f(X) - \varOmega } \right|} \hfill & {{\text{if }}\;f(X) \ge 0\;{\text{or}}\; \, res(X) = 0} \hfill \\ \end{array} } \right.$$

(16)

$$where{\kern 1pt} \,\alpha = \left\{ {\begin{array}{*{20}l} {\frac{{\left| {f(X) - \varOmega } \right|.{\kern 1pt} {{(6\sqrt {3 - 2} )} \mathord{\left/ {\vphantom {{(6\sqrt {3 - 2} )} {6\sqrt 3 }}} \right. \kern-0pt} {6\sqrt 3 }} - res(X)}}{{\left| {f(X) - \varOmega } \right| - res(X)}},} \hfill & {{\text{if}}\,f(X) \le \varOmega {\kern 1pt} ,\;res(X) < \frac{{\left| {f(X) - \varOmega } \right|}}{3}} \hfill \\ {1 - \frac{1}{{2\sqrt {{{\left| {f(X) - \varOmega } \right|} \mathord{\left/ {\vphantom {{\left| {f(X) - \varOmega } \right|} {res(X)}}} \right. \kern-0pt} {res(X)}}} }}} \hfill & {{\text{if}}\;f(X) > \varOmega ,\;{\text{if}}\;\frac{{\left| {f(X) - \varOmega } \right|}}{3}{\kern 1pt} \le res(X) \le \left| {f(X) - \varOmega } \right|} \hfill \\ {\frac{1}{2}\sqrt {\frac{{\left| {f(X) - \varOmega } \right|}}{res(X)}} ,} \hfill & {{\text{if}}\;f(X) > \varOmega ,\;res(X) > \left| {f(X) - \varOmega } \right|} \hfill \\ 0 \hfill & {{\text{if}}\;f(X) \le \varOmega } \hfill \\ \end{array} } \right.$$

(17)

4 The proposed FA-MAL and EOPM algorithms

Based on above explanation of two penalty functions, oracle method approach and modified augmented Lagrangian function approach are incorporated into FA. Flowcharts of MAL-FA and FA-EOPM are given in Figs. 2 and 3. FA is superior than GA, PSO in terms of efficiency and converging rate as shown in Yang (2009).

Fig. 2

Flow chart of proposed MAL-FA

Fig. 3

Flow chart of proposed FA-EOPM algorithm

5 Experimental results and analysis

This section contains the validation of the implementation of the proposed algorithms such as MAL-FA and FA-EOPM that are evaluated with existing population-based algorithms for solving the non-linear (constrained) optimization problems. The proposed techniques are applied to thirteen well known non-linear problems (g1–g13) (Michalewicz and Schoenauer 1996; Runarsson and Yao 2000) as shown in Table 1.

Table 1 Characteristic of thirteen benchmark functions

The characteristics of thirteen benchmark constrained functions are an objective function ($f(X)$), dimension or number of variables (D), linear inequality (LI), non-equality (NI), linear equality (LE), non-linear equality (NE) and ρ expressed the percentage (ratio of feasible value to constrained problem). For sake of convenience, all equality constrained functions $h_{j} (\overrightarrow {x} ) = 0$ are transformed into inequality constrained using $h_{j} (\overrightarrow {x} ) - \xi \le 0\quad {\text{where }}\xi = 10^{ - 4}$ (it used for number of function evacuations (NFEs) to achieve the fixed accuracy level) (Liang et al. 2006). The following parameters are confirmed for experiments of MAL-FA and FA-EOPM as shown in Table 2. The proposed algorithms run for each case has independent 30 runs, for each problem. The constrained benchmark problems are experimented on Matlab 7.0.

Table 2 Parameters used for experiments of MAL-FA and FA-EOPM algorithms

5.1 Experimental result of MAL-FA and FA-EOPM

The numerical results of MAL-FA and FA-EOPM are contained in Table 3, where global optimum value, best optimal value, mean value, worst value and standard deviation (SD) are obtained for thirteen cost function (objective value) values over 25 independent runs. The outcomes in bold indicate best solution found so far.

Table 3 Results obtained by FA-EOPM and MAL-FA on thirteen non-linear problems

As shown in Table 3, the outcomes of proposed algorithms, FA-EOPM and MAL-FA are calculated. These outcomes estimated in terms of the best optimal value, worst value, mean value and SD solutions. These experiments carried on thirteen constrained benchmark functions, which are shows good outcomes for all test functions.

MAL-FA algorithm is able to discover the well-known feasible solution consistently on test problems over 25 runs except for g02, g03, and g10 problems. As the FA has fast convergence rate, robustness and comparison of MAL-FA and FA-EOPM, MAL-FA algorithm shows exceptionally better performance for all benchmark functions.

5.2 Comparison with other state of art-algorithms of augmented Lagrangian method

To verify efficiency, effectiveness and performance of the MAL-FA algorithms are shown in Table 4, experimentations of the proposed MAL-FA algorithms are evaluated with the four population based algorithms, such as augmented Lagrangian fish swarm based method (ALFS) (Rocha et al. 2011), augmented Lagrangian ant colony optimization-based method (ALACO) (Mahdavi and Shiri 2015), hybrid genetic pattern search augmented Lagrangian method (HGPSAL) (Costa et al. 2012) and genetic algorithm based augmented Lagrangian (GAAL) (Deb and Srivastava 2012). Note that NF indicates no feasible solution so far found.

Table 4 Result obtained by MAL-FA and various metaheuristic algorithms on thirteen test problems

As shown in Table 4, after 25 times independent and successful runs of the proposed algorithm (MAL-FA), a best optimal solution along with mean value for thirteen benchmark functions was obtained. After examining the outcomes, a unique aspect is that all the algorithms in experiments found the best global feasible solutions for the benchmark functions g08 and g09.

In Table 4, comparing MAL-FA with GAAL algorithm, we can determine the best optimal value and mean value in three benchmark functions: g01, g07, and g08. In four test problems g05, g07, g12 and g13, MAL-FA algorithm is superior to GAAL algorithm in finding the best and mean outcomes. MAL-FA evaluated with ALFS algorithm, it is observed that MAL-FA algorithm contains better outcomes for nine benchmark function problems (g01, g02, g03, g04, g05, g06, g07, g10, and g11). When comparing ALACO with proposed algorithm MAL-FA, it seen that MAL-FA algorithm gives superior outcomes in four benchmark functions (g05, g10, g11 and g12), Compared with the HGPSAL algorithm, MAL-FA gives better statistical results in four benchmarks function (g01, g02, g05 and g10).

MAL-FA algorithm shows better outcomes for 13 benchmark functions due to FA having fast convergence rates and being able to automatically divide individuals (populations) into subgroups (Fister et al. 2013).

5.3 Statistical analysis

Metaheuristic algorithms are stochastic in nature, so that statistical analysis should be analyzed (Derrac et al. 2011). The best value, mean value and SD only compare the overall performance of the bio-inspired metaheuristic algorithms, while in this work, statistical analysis applied over on thirteen benchmark functions and provides the results are statistically significant. The non-parametric (i.e. Wilcoxon rank-sum) test, that can be used to verify if two sets of population are different statistically significant or not. Technically speaking, this statistical test returns a parameter called p values. In this work, an algorithm is statistically significant if and only if it results in a p value less than 0.05. The calculated values in the Wilcoxon’s test comparison of proposed MAL-FA algorithm with various metaheuristic algorithms on thirteen test problems shown in Table 5. A p value determines the significance level of two algorithms and not significance value indicate in bold in Table 5.

Table 5 p values of the Wilcoxon rank-sum test comparing MAL-FA with various metaheuristic algorithms over thirteen benchmark functions (with p ≤ 0.05)

In Table 5, it is have observed that by conducting Wilcoxon rank-sum test of MAL-FA algorithm versus various metaheuristic algorithms over on thirteen constraints test problems. Functions g01, g02, g06, g07, g12 and g13 have significance p value which less than (p ≤ 0.05). By comparing MAL-FA versus ALFS, it indicate that g03, g09 and g10 functions not have significance p value. At in MAL-FA versus ALACO g04, g05, g09 and g11 functions not have significance p value. In MAL-FA versus HGPSAL, g03 and g05 functions not have significance p value. In MAL-FA versus GAAL, g05 function not have significance p value.

6 Engineering design problems

Typically, almost all engineering design optimization problems are nonlinear which contain complex constraints. In most instances, the feasible solutions of problems do not exist. For calculating the experimental outcomes and efficiency of proposed MAL-FA algorithm, it is applied on four well-studied engineering problems that are widely used in the literature. The detailed mathematical representations of all four engineering benchmark problems designs are provided in an “Appendix”.

6.1 Rolling element bearing design problem

In this problem, the dynamic load carrying capacity (volume) of a rolling element bearing is maximized. It contains ten decision variables which have ball diameter ($D_{b}$), pitch diameter ($D_{m}$), the number of balls (Z), inner and outer raceway curvature coefficients (Chakraborty et al. 2003).

The rolling element bearing problem is optimized using ABC (Karaboga and Basturk 2007), TLBO (Rao et al. 2011), GA4 (Gupta et al. 2007) and MAL-FA. Table 5 shows the experimental outcomes obtained in term of the best optimum outcome, worst value, mean value, SD and NFEs.

As given in Table 6, by evaluating of NFEs for MAL-FA algorithm, it can be seen that MAL-FA algorithm is superior than other metaheuristic algorithms. In Fig. 4, shows a graph plot of the proposed algorithm MAL-FA in comparison with other metaheuristic algorithms from the literature with respect to NFEs.

Table 6 Experimental outcomes from metaheuristic algorithms for rolling element bearing problem

Fig. 4

Comparison of metaheuristic algorithms with NFEs for rolling element bearing problem

6.2 Welded beam problem

For a substantial minimization of substantially cost of fabrication, MAL-FA algorithm is tested for a non-linear welded beam problem. The MAL-FA algorithm is compared with GA + APM (Lemonge and Barbosa 2004), GA-AIS (Bernardino et al. 2008), SSaDE (Huang et al. 2006), DUVDE-APM (da Silva et al. 2011) GA2 (Coello 2000), GA3 (Coello and Montes 2002), CAEP (Coello Coello and Becerra 2004), MGA (Coello Coello 2000), CPSO (Parsopoulos and Vrahatis 2002) and WCA (Eskandar et al. 2012). The comparisons of statistical results are shown in Table 7. The experimental outcomes obtained by the MAL-FA outperformed the obtained results in term of best value and NFEs.

Table 7 Experimental outcomes from metaheuristic algorithms for welded beam problem

As given in Table 7, the best optimum value of proposed algorithm MAL-FA converges to the minimum value of welded design beam problem and a value obtained from the proposed algorithm is better than other optimization algorithms given in Fig. 5. Comparison of MAL-FA with some metaheuristic algorithms with the NFEs shows better result value for welded beam problem as given in Fig. 6.

Fig. 5

Comparison of best values of metaheuristic algorithms for welded beam problem

Fig. 6

Comparison of metaheuristic algorithms with NFEs for welded beam problem

6.3 Multiple disk clutch brake problem

It is a discrete minimization problem and the main task is the minimization of the mass of multiple disc clutch brakes using five different design (dimensional) variables: inner radius, outer radius, thickness of the disc, actuating force and number of friction surfaces.

It is optimized using ABC, TLBO, WCA (water cycle algorithm) and MAL-FA. As shown in Table 8, shows the experimental outcomes. The result obtained by MAL-FA has outperformed other results with respect to best solution and function evaluation.

Table 8 Experimental outcomes from metaheuristic algorithms for multiple disk clutch brake design problems

As given in Table 8, the MAL-FA algorithm detected the best optimum outcomes with considerable improvement for this problem compared to other optimization algorithms as shown in Fig. 7. In terms of experimental outcomes for this problem, MAL-FA obtained better outcomes using less NFEs compared to the metaheuristic algorithms as shown in Fig. 8.

Fig. 7

Best value of metaheuristic algorithms for multiple disk clutch brake design problem

Fig. 8

Comparison of metaheuristic algorithms with NFEs for multiple disk clutch brake problems

6.4 Three bar truss problem

It is one of the engineering design problems which contains structural parameters and loading. The evaluation of the statistical outcomes of the MAL-FA proposed algorithm along with the SC (Ray and Liew 2003), HEAA (Wang et al. 2009), DEDS (Zhang et al. 2008), and PSO–DE (Mezura-Montes et al. 2006) are shown in Table 9. The experimental outcomes of MAL-FA shows superior outcomes in terms of NFEs against other metaheuristic algorithms as shown in Fig. 9.

Table 9 Experimental outcomes from metaheuristic algorithms for three-bar truss problem

Fig. 9

Comparison of metaheuristic algorithms with NFEs for three-bar truss design problem

It can be seen that experimental outcomes for engineering problems of MAL-FA algorithm with respect to other metaheuristic algorithms (such as PSO, GA. etc.) are better because it avoids premature convergence and has fast convergence rate.

7 Conclusions and future work

This paper presents two new constrained optimization algorithms such as the oracle penalty method and the augmented Lagrangian method incorporated into the firefly algorithm. These two new algorithms (1) firefly algorithm with extended oracle penalty method (FA-EOPM) and (2) modified augmented Lagrangian with firefly algorithm (MAL-FA) are proposed due to various advantages of FA algorithm such as fast convergence rate given by randomness parameter alpha (α) and, dividing population (fireflies) into subgroups. The implementation of our proposed algorithms has been applied on thirteen constrained functions and evaluated in comparison with several widely used population-based constrained optimization algorithms. The outcomes obtained by these algorithms show better performance, efficiency and effectiveness compared to other metaheuristic algorithms. Using statistical analysis, MAL-FA algorithm shows effective outcomes against various algorithms in term of NFEs.

It was observed that the MAL-FA finds better outcomes on specific problems compared to FA-EOPM. Also, the MAL-FA algorithm can be used to solve engineering design problems efficiently. In the present work, we use improved FA method to solve the engineering optimization problem. In future, we can use some other metaheuristic algorithms to solve this problem, such as monarch butterfly optimization (MBO), earthworm optimization algorithm (EWA), elephant herding optimization (EHO) and moth search (MS) algorithm.

Appendix

1.1 Rolling element bearing design problem

$$\begin{aligned} & { \hbox{max} }\;C_{d} = f_{{_{c} }} Z^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} D_{b}^{1.8} \quad {\text{if }}D_{b} \le 25.4\,{\text{mm}} \\ & C_{d} = 3.647Z^{{{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} D_{b}^{1.4} \quad {\text{if }}D_{b} \succ 25.4\,{\text{mm}} \\ & {\text{subject to:}} \\ & {\text{g}}_{1} \left( X \right) = \frac{{\phi_{0} }}{{2\sin^{ - 1} \left( {{\raise0.7ex\hbox{${D_{b} }$} \!\mathord{\left/ {\vphantom {{D_{b} } {D_{m} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${D_{m} }$}}} \right)}} - Z + 1 \ge 0 \\ & {\text{g}}_{2} \left( X \right) = 2D_{b} - KD_{\hbox{min} } \left( {D - d} \right) \ge 0 \\ & {\text{g}}_{4} \left( X \right) = \xi B_{w} - D_{b} \ge 0 \\ & {\text{g}}_{3} \left( X \right) = K_{D\hbox{max} } \left( {D - d} \right) - 2D_{b} \ge 0 \\ & {\text{g}}_{5} \left( X \right) = D_{m} - 0.5\left( {D + d} \right) \ge 0 \\ & {\text{g}}_{6} \left( X \right) = \left( {0.5 + e} \right)\left( {D + d} \right) - D_{m} \ge 0 \\ & {\text{g}}_{7} \left( X \right) = 0.5\left( {D - D_{m} - D_{b} } \right) - \xi D_{b} \ge 0 \\ & {\text{g}}_{8} \left( X \right) = f_{1} \ge 0.515 \\ & {\text{g}}_{9} \left( X \right) = f_{0} \ge 0.515 \\ & {\text{where }} \\ & f_{c} = 37.91 + \left[ {1 + \left\{ {1.04\left( {\frac{1 - \gamma }{1 + \gamma }} \right)^{1.72} \left( {\frac{{f_{1} \left( {2f_{0} - 1} \right)}}{{f_{0} \left( {2f_{1} - 1} \right)}}} \right)^{0.41} } \right\}^{{{\raise0.7ex\hbox{${10}$} \!\mathord{\left/ {\vphantom {{10} 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} } \right]^{ - 0.3} \\ & \quad \quad \times \left[ {\frac{{\gamma^{0.3} \left( {1 - \gamma } \right)^{1.39} }}{{(1 + \gamma )^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} }}} \right]\left[ {\frac{{2f_{i} }}{{2f_{i} - 1}}} \right]^{0.41} \\ & \phi_{0} = 2\varPi - 2 \times \cos^{ - 1} \left( {\frac{{\left[ {\left\{ {{{\left( {D - d} \right)} \mathord{\left/ {\vphantom {{\left( {D - d} \right)} {2 - 3\left( {{T \mathord{\left/ {\vphantom {T 4}} \right. \kern-0pt} 4}} \right)^{2} + \left\{ {{D \mathord{\left/ {\vphantom {D {2 - {T \mathord{\left/ {\vphantom {T {4 - D_{b} }}} \right. \kern-0pt} {4 - D_{b} }}}}} \right. \kern-0pt} {2 - {T \mathord{\left/ {\vphantom {T {4 - D_{b} }}} \right. \kern-0pt} {4 - D_{b} }}}}} \right\}^{2} - \left\{ {{d \mathord{\left/ {\vphantom {d {2 + {T \mathord{\left/ {\vphantom {T 4}} \right. \kern-0pt} 4}}}} \right. \kern-0pt} {2 + {T \mathord{\left/ {\vphantom {T 4}} \right. \kern-0pt} 4}}}} \right\}^{2} }}} \right. \kern-0pt} {2 - 3\left( {{T \mathord{\left/ {\vphantom {T 4}} \right. \kern-0pt} 4}} \right)^{2} + \left\{ {{D \mathord{\left/ {\vphantom {D {2 - {T \mathord{\left/ {\vphantom {T {4 - D_{b} }}} \right. \kern-0pt} {4 - D_{b} }}}}} \right. \kern-0pt} {2 - {T \mathord{\left/ {\vphantom {T {4 - D_{b} }}} \right. \kern-0pt} {4 - D_{b} }}}}} \right\}^{2} - \left\{ {{d \mathord{\left/ {\vphantom {d {2 + {T \mathord{\left/ {\vphantom {T 4}} \right. \kern-0pt} 4}}}} \right. \kern-0pt} {2 + {T \mathord{\left/ {\vphantom {T 4}} \right. \kern-0pt} 4}}}} \right\}^{2} }}} \right\}} \right]}}{{2\left\{ {{{\left( {D - d} \right)} \mathord{\left/ {\vphantom {{\left( {D - d} \right)} 2}} \right. \kern-0pt} 2}} \right\} - 3\left( {{T \mathord{\left/ {\vphantom {T 4}} \right. \kern-0pt} 4}} \right)^{2} + \left\{ {{D \mathord{\left/ {\vphantom {D {2 - {T \mathord{\left/ {\vphantom {T {4 - D_{b} }}} \right. \kern-0pt} {4 - D_{b} }}}}} \right. \kern-0pt} {2 - {T \mathord{\left/ {\vphantom {T {4 - D_{b} }}} \right. \kern-0pt} {4 - D_{b} }}}}} \right\}}}} \right) \\ & \gamma = \frac{{D_{b} \cos \alpha }}{{D_{m} }},\;f_{1} = \frac{{r_{1} }}{{D_{b} }},\;f_{0} = \frac{{r_{0} }}{{D_{b} }},\;T = D - d - 2D_{b} \\ & D = 160,\;d = 90,\;B_{w} = 30,\;r_{i} = r_{0} = 11.033 \\ & 0.5\left( {D + d} \right) \le D_{m} \le 0.6\left( {D + d} \right),\;0.15\left( {D + d} \right) \le D_{m} \le 0.45\left( {D + d} \right), \\ & 0.515 \le f_{i} ,\;f_{0} \le 0.6,\;0.4 \le K_{D\hbox{min} } \le 0.5,\;0.6 \le K_{D\hbox{max} } \le 0.7,\;0.3 \le \varepsilon \le 0.4, \\ & 0.02 \le e \le 0.01,\;0.6 \le \xi \le 0.85,\;4 \le Z \le 50 \\ \end{aligned}$$

1.2 Welded beam design problem

$$\begin{aligned} & \hbox{min} \;f\left( X \right) = 1.10471x_{1}^{2} x_{2} + 0.04811x_{3} x_{4} \left( {14 + x_{2} } \right) \\ & {\text{subject to:}} \\ & {\text{g}}_{1} \left( X \right) = \tau \left( {\overrightarrow {x} } \right) - \tau_{\hbox{max} } \le 0 \\ & {\text{g}}_{2} \left( X \right) = \sigma \left( {\overrightarrow {x} } \right) - \sigma_{\hbox{max} } \le 0 \\ & {\text{g}}_{3} \left( X \right) = x_{1} - x_{4} \le 0 \\ & {\text{g}}_{4} \left( X \right) = 0.10471x_{1}^{2} + 0.4811x_{3} x_{4} \left( {14 + x_{2} } \right) - 5 \le 0 \\ & {\text{g}}_{5} \left( X \right) = 0.125 - x_{1} \le 0 \\ & {\text{g}}_{6} \left( X \right) = \delta \left( {\overrightarrow {x} } \right) - \delta_{\hbox{max} } \le 0 \\ & g_{7} \left( X \right) = P - P_{c} \left( {\overrightarrow {x} } \right) \le 0 \\ & 0.1 \le x_{i} \le 2\quad i = 1,4 \\ & 0.1 \le x_{i} \le 10\quad i = 2,3 \\ & {\text{where}} \\ & \tau \left( {\overrightarrow {x} } \right) = \sqrt {\left( {\tau^{\iota } } \right) + 2\tau^{\iota } \tau^{\iota \iota } \frac{{x_{2} }}{2R}\tau^{\iota \iota } } ,\quad \tau^{\iota } = \frac{P}{{\sqrt 2 x_{1} x_{2} }},\quad \tau^{\iota \iota } = \frac{MR}{J} \\ & M = P\left( {L + \frac{{x_{2} }}{2}} \right),\quad R = \sqrt {\frac{{x_{2}^{2} }}{4} + \left( {\frac{{x_{1} + x_{2} }}{2}} \right)^{2} } \\ & J = 2\left\{ {\sqrt 2 x_{1} x_{2} \left[ {\sqrt {\frac{{x_{2}^{2} }}{4} + \left( {\frac{{x_{1} + x_{2} }}{2}} \right)^{2} } } \right]} \right\},\quad \sigma \left( {\overrightarrow {x} } \right) = \frac{6PL}{{x_{4} x_{3}^{2} }}, \\ & \delta \left( {\overrightarrow {x} } \right) = \frac{{4PL^{3} }}{{Ex_{3}^{3} x_{4} }},\quad P_{c} \left( {\overrightarrow {x} } \right) = \frac{{4.013E\sqrt {\frac{{x_{3}^{2} x_{4}^{6} }}{36}} }}{{L^{2} }}\left( {1 - \frac{{x_{3} }}{2L}\sqrt {\frac{E}{4G}} } \right) \\ & P = 6000\,{\text{lb}},\;L = 14\,{\text{in}},\;E = 30 \times 10^{6} \,{\text{psi}},\;G = 12 \times 10^{6} \,{\text{psi}} \\ & \tau_{\hbox{max} } = 13,600\,{\text{psi}},\;\sigma_{\hbox{max} } = 30,000\,{\text{psi}},\;\delta_{\hbox{max} } = 0.25\,{\text{in}} \\ \end{aligned}$$

1.3 Multiple disk clutch brake design problem

$$\begin{aligned} & \hbox{min} \;f\left( X \right) = \pi \left( {r_{0}^{2} - r_{1}^{2} } \right)t\left( {Z + 1} \right)\rho \\ & g_{1} \left( X \right) = r_{0} - r_{i} - \Delta r \ge 0 \\ & g_{2} \left( X \right) = l_{\hbox{max} } - \left( {Z + 1} \right)\left( {t + \delta } \right) \ge 0 \\ & g_{3} \left( X \right) = p_{\hbox{max} } - p_{rz} \ge 0 \\ & g_{4} \left( X \right) = p_{\hbox{max} } v_{sr\hbox{max} } - p_{rz} v_{sr} \ge 0 \\ & g_{5} \left( X \right) = v_{sr\hbox{max} } - v_{sr} \ge 0 \\ & g_{6} \left( X \right) = T_{\hbox{max} } - T \ge 0 \\ & g_{7} \left( X \right) = M_{h} - sM_{s} \ge 0 \\ & g_{8} \left( X \right) = T \ge 0 \\ & M_{h} = \frac{2}{3}\mu FZ\frac{{r_{0}^{3} - r_{i}^{3} }}{{r_{0}^{2} - r_{i}^{2} }},\quad p_{rz} = \frac{F}{{\pi \left( {r_{0}^{2} - r_{i}^{2} } \right)}}, \\ & v_{sr} = \frac{{2\pi n\left( {r_{0}^{3} - r_{i}^{3} } \right)}}{{90\left( {r_{0}^{2} - r_{i}^{2} } \right)}},\quad T = \frac{{I_{z} \pi n}}{{30\left( {M_{h} + M_{f} } \right)}} \\ & \Delta = 20\,{\text{mm}},\,t_{\hbox{max} } = 3\,{\text{mm}},\;f_{\hbox{max} } = 1.5\,{\text{mm}},\;\mu = 0.5, \\ & s = 1.5,\;M_{s} = 40\,{\text{nm}},M_{f} = 3\,{\text{Nm}},\;n = 250\,{\text{rpm,}} \\ & p_{\hbox{max} } = 1\,{\text{Mpa,}}\;I_{z} = 55\,{\text{kg mm}}^{3} ,\,\,T_{\hbox{max} } = 15\,{\text{s,}} \\ & F_{\hbox{max} } = 1000\,{\text{N,}}\,\,r_{\hbox{min} } = 55\,{\text{mm}},\;r_{0} = 110\,{\text{mm}} \\ \end{aligned}$$

1.4 Three bar truss design problem

$$\begin{aligned} & f\left( X \right) = \left( {2 + \sqrt 2 x_{1} x_{2} } \right) \times l \\ & {\text{subject to:}} \\ & g_{1} \left( X \right) = \frac{{\sqrt 2 x_{1} + x_{2} }}{{\sqrt 2 x_{1}^{2} + 2x_{1} x_{2} }}p - \rho \le 0 \\ & g_{2} \left( X \right) = \frac{{x_{2} }}{{\sqrt 2 x_{1}^{2} + 2x_{1} x_{2} }}p - \rho \le 0 \\ & g_{1} \left( X \right) = \frac{1}{{\sqrt 2 x_{1}^{2} + 2x_{1} x_{2} }}p - \rho \le 0 \\ & 0 \le x_{1} \le 1\quad i = 1,2 \\ & l = 100\,{\text{cm}},P = 2\,{\text{kN}}/{\text{cm}}^{ 2} ,\sigma = 2\,{\text{kN}}/{\text{cm}}^{ 3} \\ \end{aligned}$$