Optimum synthesis of a four-bar mechanism using the modified bacterial foraging algorithm

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Optimum synthesis of a four-bar mechanism using themodified bacterial foraging algorithmEfrén Mezura-Montes a , Edgar A. Portilla-Flores b & Betania Hernández-Ocaña aa Laboratorio Nacional de Informática Avanzada (LANIA) A.C., Rébsamen 80, Xalapa, Mexicob Centro de Innovación y Desarrollo Tecnológico en Cómputo (CIDETEC-IPN), U. Adolfo LópezMateos, México D.F., MexicoPublished online: 18 Jan 2013.

To cite this article: Efrén Mezura-Montes , Edgar A. Portilla-Flores & Betania Hernández-Ocaña (2013): Optimum synthesisof a four-bar mechanism using the modified bacterial foraging algorithm, International Journal of Systems Science,DOI:10.1080/00207721.2012.745023

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International Journal of Systems Science, 2012http://dx.doi.org/10.1080/00207721.2012.745023

Optimum synthesis of a four-bar mechanism using the modified bacterial foraging algorithm

Efren Mezura-Montesa,∗, Edgar A. Portilla-Floresb and Betania Hernandez-Ocanaa

aLaboratorio Nacional de Informatica Avanzada (LANIA) A.C., Rebsamen 80, Xalapa, Mexico; bCentro de Innovacion y DesarrolloTecnologico en Computo (CIDETEC-IPN), U. Adolfo Lopez Mateos, Mexico D.F., Mexico

(Received 26 September 2011; final version received 28 August 2012)

This paper presents the mechanical synthesis of a four-bar mechanism, its definition as a constrained optimisation problemin the presence of one dynamic constraint and its solution with a swarm intelligence algorithm based on the bacteria foragingprocess. The algorithm is adapted to solve the optimisation problem by adding a suitable constraint-handling technique thatis able to incorporate a selection criterion for the two objectives stated by the kinematic analysis of the problem. Moreover,a diversity mechanism, coupled with the attractor operator used by bacteria, is designed to favour the exploration of thesearch space. Four experiments are designed to validate the proposed model and to test the performance of the algorithmregarding constraint-satisfaction, sub-optimal solutions obtained, performance metrics and an analysis of the solutions basedon the simulation of the four-bar mechanism. The results are compared with those provided by four algorithms found inthe specialised literature used to solve mechanical design problems. On the basis of the simulation analysis, the solutionsobtained by the proposed algorithm lead to a more suitable design based on motion generation and operation quality.

Keywords: optimum synthesis; four-bar mechanism; nature-inspired optimisation; swarm intelligence; evolutionaryalgorithms

1. Introduction

Nowadays, in the engineering design framework, one ofthe most important approaches is the usage of computer-aided design software. This approach allows the designengineer to obtain virtual three-dimensional models, two-dimensional drawings and numerical control code, amongother features. However, the design of modern systemscould be complex. One design approach commonly foundin the specialised literature is the mechatronic approach,where the design of modern systems can be consideredas a mechatronic design. This approach requires that amechanical system and its control system are designed asan integrated one; then a concurrent design problem is pro-posed in order to obtain the whole system (Portilla-Flores,Mezura-Montes, Alvarez Gallegos, Coello Coello, andCruz-Villar 2007). An alternative design approach consid-ers assigning the best possible combination of values to aset of parameters that describe the system. In this approach,a kinematic analysis is carried out in the first stage. Thekinematic analysis is used to fulfil positions and velocitiesof the mechanical system, and it also allows the system tobe described by a set of parameters. Moreover, the designengineer must propose a set of performance functions andconstraints to quantify the system behaviour. Thereafter,the design engineer is able to propose several potentialsolutions (Norton 1997). Precisely in this part of the designapproach, the kinematic analysis can be transformed into

∗Corresponding author. Email: [email protected]

an optimisation problem and then solved by an optimisationmethod (Nariman-Zadeh, Felezi, Jamali, and Ganji 2009;Radovan and Stevan 2009; Saridakis and Dentsoras 2009).

Mathematical programming, such as the Newtonmethod, has been adopted as in the work by Mermetas(2004) where an optimal kinematic design of a planarmanipulator with four-bar mechanism was optimised.There, optimum link measurements of the manipulatorthat maximise the local mobility index depending on theinput link location were found. Also, design charts for theoptimum manipulator design were obtained.

Nonetheless, based on the inherent difficulty of theoptimisation problems obtained from the real-world systembehaviour of mechanical systems, it may be difficult tosolve them with traditional mathematical programmingmethods. Therefore, nature-inspired meta-heuristic algo-rithms have become a valid option. They can be dividedin two main classes (Engelbrecht 2007): (1) evolutionaryalgorithms (EAs) and (2) swarm intelligence algorithms(SIAs). The first ones emulate the evolution of speciesand the survival of the fittest. The second ones emulatethe social behaviour of some simple species searching forfood or shelter. EAs comprise different algorithms suchas genetic algorithms (GAs) (Eiben and Smith 2003),evolution strategies (Schwefel 1995), evolutionary pro-gramming (Fogel 1999), genetic programming (Koza et al.2003) and differential evolution (DE) (Price, Storn, and

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http://dx.doi.org/10.1080/00207721.2012.745023

2 E. Mezura-Montes et al.

Lampinen 2005). SIAs’ main approaches are the particleswarm optimisation (PSO) (Kennedy and Eberhart 2001)and the ant colony optimisation (ACO) (Dorigo, Maniezzo,and Colorni 1996). PSO is based on the choreographiesfollowed by birds and is able to solve difficult numericaloptimisation problems (Daneshyari and Yen 2012; Wang,Yang, Ip, and Wang 2012). ACO models different be-haviours found in ant nests but has been mainly used to solvecombinatorial optimisation problems. Among those novelSIAs, there is one that emulates the foraging behaviour ofbacteria. It was proposed by Passino (2002), and it is calledbacterial foraging optimisation algorithm (BFOA). Theentire foraging process of the Escherichia coli bacteriumis emulated: Chemotaxis (tumble-swim movements),reproduction and elimination-dispersal. More recently, themodified BFOA (MBFOA) was proposed by Mezura-Montes and Hernandez-Ocana (2009) with the aim toincrease its applicability in engineering design by reducingthe number of user-defined parameters and handlingconstraints as well.

The following are approaches based on EAs for op-timising mechanical systems. In Mundo and Yan (2007),a method for the kinematic optimisation of transmissionmechanism was proposed. A motion control of a ball-screwtransmission mechanism was developed. The kinematiccharacteristics of the ball-screw mechanism were analysedby means of non-dimensional motion equations in orderto formulate an optimisation problem. A GA was imple-mented in order to optimise the objective function anda penalty method was used to fulfil the design rules. InNariman-Zadeh, Felezi, Jamali, and Ganji (2009), a hybridmulti-objective GAs was used for Pareto optimum synthe-sis of the four-bar linkages considering the minimisationof two objective functions simultaneously. The obtainedPareto fronts demonstrated that trade-offs between thesetwo objectives can be recognised so that a designer can se-lect a desired four-bar linkage from the set of sub-optimalsolutions. In Radovan and Stevan (2009), the synthesis ofa four-bar linkage in which the coupler point performsapproximately a rectilinear motion was presented. Veryhigh accuracy for motion along a straight line at a largenumber of given points was achieved by using the vari-able controlled deviations method and by also employingDE.

From the previously commented works, it was notedthat (1) different search algorithms, e.g. GAs and DE, havebeen adapted to obtain the optimal design synthesis of four-bar mechanisms and (2) each mechanical system providesnew and specific design challenges. These two findings arethe main motivation of this paper.

The objective of the present work is twofold: (1) tooptimise the output motion and energy transmission of acontinuously variable transmission (CVT) by designinga four-bar mechanism on the crank–rocker configuration(both explained later in the paper) and (2) to extend

the solution of this type of optimal mechanical designproblems to other meta-heuristic (MBFOA) not yet applied,to the best of the authors’ knowledge.

As it will be detailed later in the paper, the optimi-sation problem obtained from the mechanical system is aconstrained bi-objective optimisation problem and one ofthe constraints is dynamic. A set of experiments is carriedout to analyse the behaviour of the algorithm regarding thefeasibility and the treatment of the dynamic constraint, theoverall performance of the algorithm based on metrics and,finally, an analysis of the solutions based on simulations.Three algorithms found in the specialised literature are usedfor comparison purposes and statistical tests are applied toget confidence on the results presented.

The paper is organised as follows: Section 2 outlinesthe mechanism synthesis and the design problem of thefour-bar mechanism. A kinematic analysis of the four-barmechanism is also shown. Section 3 presents the develop-ment of the optimal strategy, i.e. the design of the objectivefunctions and the constraints. Thereafter, the bi-objectiveoptimisation problem and the design variables are detailedin Section 4. Section 5 then presents the proposed algorithmbased on MBFOA by detailing those added mechanisms soas to get it suitable to solve the bi-objective optimisationproblem in the presence of a dynamic constraint. After that,the experimental design, the obtained results and their cor-responding discussions are included in Section 6. Finally,some conclusions and the future work are established inSection 7.

2. Mechanism synthesis problem

There are several mechanisms using pedalling motion as asupply source for machinery. However, these mechanismshave some inherent difficulties due to the complexity ofbuilding a pinion–chain–crown mechanism. In Pardo andSanchez (2006), a new CVT configuration was proposed.It was based on a crank–slider mechanism to use pedallingmotion. The operational principle of this mechanism isexplained as follows: the mechanism converts the inputpedalling motion onto the driving wheel rotating input byunilateral transmission. To do this, the mechanism usesratchets that are alternatively engaged inside the linksof the output chain to push and pull them. On the otherhand, a special mechanism, with the aim to change theeffective radius of pulling in the whole mechanical system,is considered. The above-mentioned CVT is proposed withthe goal to introduce a gradual, linear change of velocityon pedaling vehicles. A simplified drawing from thismechanism can be seen in Figure 1.

As it can be observed, in the above-mentioned CVTconfiguration, the special mechanism used for convertingthe input pedalling motion is very difficult to build. That isdue to the particular manufacturing of the shaft that cou-ples the input pedalling motion into the rest of the CVT.

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International Journal of Systems Science 3

Figure 1. Traditional continuously variable transmission.

Therefore, in this paper, a four-bar mechanism is used toreplace such mechanism. It is important to remark that theselection of such mechanism is based on the required kine-matic by the whole system and the fact that the four-barmechanisms have been widely studied and they are easy tobuild. A drawing from the CVT configuration proposed inthis paper can be observed in Figure 2.

An important fact is that the energy transmission fromthe input to the mechanical system in Figure 2 is related tothe rocker motion amplitude of the four-bar mechanism. Inother words, the input motion of the crank–slider mecha-nism is the rocker motion of the first mechanism. On theother hand, it is known that the velocity of the slider is amathematical relationship of the ‘sin’ of the crank angle(Shigley and Uicker 1995), i.e. the larger the value of thisangle, the larger the velocity on the slider of the secondmechanism. As a matter of fact, once the range of motionis satisfied, it is necessary to ensure a smooth transmis-sion of force and speed on the joint of the connecting rodand the rocker of the four-bar mechanism. Therefore, get-ting mechanical elements, which allow a large amplitudeon the motion of the rocker, is a goal of the mechanicaldesign.

When the design engineer carries out the design ofmechanisms, both analysis and synthesis are taken into ac-count. Regarding dimensional synthesis, there are threeproblems to be considered: function generation, pathgeneration and motion generation. In a function generationproblem, an input motion is correlated with an output mo-tion on a mechanism. The path generation problem is relatedto the control of a point within a plane, so that it can fol-low a path or trajectory previously established. Finally, amotion generation problem is defined by the control of aline within a plane, so that a set of sequential positions isfulfilled. The last problem is more general with respect tothe path generation problem (Norton 1997).

In this paper, the design of the four-bar mechanismprecisely belongs to the motion generation problem. There-fore, two approaches are used to carry out the design of thefour-bar mechanism: a dimensional synthesis for motiongeneration and the performance of the mechanism.

2.1. Kinematic analysis of the four-barmechanism

As it has been previously mentioned, the CVT is basedon two mechanisms: one four-bar mechanism and onecrank–slider mechanism. A kinematic analysis of the firstmechanical system is carried out in order to obtain thebehaviour of the whole system. A schematic drawing ofthe four-bar mechanism is shown in Figure 3.

This four-bar mechanism is composed of a referencebar (r1), a crank bar (r2), a connecting rod bar (r3) anda rocker bar (r4). A set of four important angles has tobe considered, beginning with θ1, described by the anglebetween the horizontal axis and r1, θ2 is the angle between

Figure 2. Continuously variable transmission proposed in this paper.

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Figure 3. Four-bar mechanism diagram.

the horizontal axis and r2, θ3 is the angle between thehorizontal axis and r3, and θ4 is the angle between the hori-zontal axis and r4. The angle marked as μ is the transmissionangle. This special configuration is the so-called crank–rocker.

2.1.1. Analysis of position

For the mechanism in Figure 3, a loop-closure equation(Shigley and Uicker 1995) is proposed as follows:

�r1 + �r4 = �r2 + �r3, (1)

where each one of the vector is related to each one of thelinkages.

On the other hand, if vectors are written in polar form,Equation (1) can be expressed as follows:

r1ejθ1 + r4e

jθ4 = r2ejθ2 + r3e

jθ3 . (2)

Applying Newton’s formula into Equation (2), it can bewritten as

r1(cosθ1 + jsin θ1) + r4(cosθ4 + jsinθ4)

= r2(cosθ2 + jsinθ2) + r3(cosθ3 + jsinθ3). (3)

Separating the real and imaginary parts of Equation (3),

r1cosθ1 + r4cosθ4 = r2cosθ2 + r3cosθ3

jr1sinθ1 + jr4sinθ4 = jr2sinθ2 + jr3sinθ3. (4)

To obtain the angular position θ3, the left-hand side of theequation system (4) is put in terms of θ4,

r4cosθ4 = r2cosθ2 + r3cosθ3 − r1cosθ1

r4sinθ4 = r2sinθ2 + r3sinθ3 − r1sinθ1. (5)

Taking the square of Equation (5) and adding them,Freudenstein’s equation (Shigley and Uicker 1995) in acompact form is established as follows:

A1cosθ3 + B1sin θ3 + C1 = 0, (6)

where

A1 = 2r3 (r2cosθ2 − r1cosθ1) (7)

B1 = 2r3 (r2sinθ2 − r1sinθ1) (8)

C1 = r21 + r2

2 + r23 − r2

4 − 2r1r2cos (θ1 − θ2) . (9)

The angle θ3 can be explicitly found as a function of theparameters A1, B1, C1 and θ2. Such solution is obtained byexpressing sinθ3 and cosθ3 in terms of tan

(θ32

)as follows:

sinθ3 = 2tan(

θ32

)1 + tan2

(θ32

) , cosθ3 = 1 − tan2(

θ32

)1 + tan2

(θ32

) (10)

and substituting those in Equation (6), a second-order linearequation is obtained,

[C1 − A1] tan2

(θ3

2

)+ [2B1] tan

(θ3

2

)+ A1 + C1 = 0.

(11)Solving Equation (11), the angular position θ3 is given byEquation (12),

θ3 = 2arctan

⎡⎣−B1 ±

√B2

1 + A21 − C2

1

C1 − A1

⎤⎦ . (12)

A similar mathematical procedure to obtain θ4 must bedone. Freudestein’s equation from Equation (1) in compactform is given by Equation (13),

D1cosθ4 + E1sinθ4 + F1 = 0, (13)

where

D1 = 2r4 (r1cosθ1 − r2cosθ2) (14)

E1 = 2r4 (r1sinθ1 − r2sinθ2) (15)

F1 = r21 + r2

2 + r24 − r2

3 − 2r1r2cos (θ1 − θ2) . (16)

Therefore, the angular position θ4 is given byEquation (17),

θ4 = 2arctan

⎡⎣−E1 ±

√D2

1 + E21 − F 2

1

F1 − D1

⎤⎦ . (17)

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Table 1. Choosing the sign of the radical according to the typeof mechanism.

Four-bar mechanism configuration θ3 θ4

Open +√ –√Crossed –√ +√

Equations (12) and (17) must take the signs of radicalsaccording to the four-bar mechanism used. Table 1 showsthe corresponding signs. It is important to remark that inthis paper, the open configuration was used.

2.1.2. Analysis of velocity

To carry out the analysis of velocity, Equation (1) is timederivative. It is important to remark that θ1 = constant,therefore,

jr4ω4ejθ4 = jr2ω2e

jθ2 + jr3ω3ejθ3 , (18)

where

ω2 = dθ2

dt(19)

ω3 = dθ3

dt(20)

ω4 = dθ4

dt. (21)

Dividing Equation (18) by jejθ3 ,

r4ω4ej (θ4−θ3) = r2ω2e

j (θ2−θ3) + r3ω3. (22)

Applying Newton’s formula into Equation (22), it can bewritten as

r4ω4[cos(θ4 − θ3) + jsin(θ4 − θ3)] = r2ω2[cos(θ2 − θ3)

+ jsin(θ2 − θ3)] + r3ω3. (23)

Taking the imaginary part of Equation (23),

r4ω4sin(θ4 − θ3) = r2ω2sin(θ2 − θ3). (24)

From Equation (24), the angular velocity ω4 can be calcu-lated as follows:

ω4 =(

r2

r4

)[sin(θ2 − θ3)

sin(θ4 − θ3)

]ω2. (25)

In this mechanism, the input engine is coupled to the shaftD and the output motion is coupled to the shaft C. That is,the input angular velocity of the mechanism is the angularvelocity of the crank (ω2) and the output angular velocity isthe angular velocity of the rocker (ω4), which is established

in Equation (25). Finally, as the magnitude of the inputangular velocity of the four-bar mechanism is constant, theangular position θ2 is given by Equation (26),

θ2 = ω2t, (26)

where ω2 = 6.9 rad/sec, which is the only parameter in themodel, is the constant speed of the input motor.

3. Optimal strategy

Once the kinematic analysis was developed, the designproblem must be established as an optimisation problem.To do this, objective function and constraints must be pro-posed.

3.1. Objective functions

Recalling from Section 2, the four-bar mechanism is de-signed by using the case of motion generation and its cor-responding performance as detailed below.

3.1.1. Motion generation

From the kinematic analysis in Section 2.1, it was notedthat the rocker motion is mechanically constrained to theshaft C. Also, the rotational displacement of this elementis called θ4 and it has the maximum and minimum valuesat the tightening points of the mechanism. The tighteningpoints are when the linkages r2 and r3 are collinear. At thispoint, one degree-of-freedom is dropped. The tighteningpoints can be observed in Figure 4(a) and 4(b).

Figure 4. Tightening points in a four-bar mechanism.

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Taking into account the tightening points, the displace-ment of the angle θ4 between these positions can be calcu-lated as indicated in Equation (27),

�1 = θ4max − θ4min, (27)

where θ4max and θ4min are computed according to the four-bar configuration as follows:

• Case (a) θ1 < 0:

θ4max = π −[abs(θ1) + arccos

(r2

1 + r24 − (r3 − r2)2

2r1r4

)](28)

θ4min = π −[abs(θ1) + arccos

(r2

1 + r24 − (r3 + r2)2

2r1r4

)]. (29)

• Case (b) θ1 = 0:

θ4max = π − arccos

(r2

1 + r24 − (r3 − r2)2

2r1r4

)(30)

θ4min = π − arccos

(r2

1 + r24 − (r3 + r2)2

2r1r4

). (31)

• Case (c) θ1 > 0:

θ4max = π +[abs(θ1) − arccos

(r2

1 + r24 − (r3 − r2)2

2r1r4

)](32)

θ4min = π +[abs(θ1) − arccos

(r2

1 + r24 − (r3 + r2)2

2r1r4

)]. (33)

It is important to remark that a maximum value for objectivefunction expressed in Equation (27) implies the maximisa-tion of the output of the four-bar mechanism.

3.1.2. Operation quality

One of the most popular criteria to evaluate the performanceof the four-bar mechanism is the transmission angle (μ). In ageneral way, the transmission angle is taken as the absolutevalue of the acute angle of the pair of angles formed atthe intersection of the connecting of the two links. Thisparameter is defined as the angle between the rocker and theconnecting rod of the four-bar mechanism. When the rockerand the connecting rod form an acute angle, μ is computedby Equation (34). If the angle between these elements is notacute, then μ = 180◦ − μ (Norton 1997). The transmissionangle varies from a minimum to a maximum value alongthe cycle of r2. Figure 5(a) and 5(b) show the positions

Figure 5. Maximum and minimum transmission angles.

of the minimum and maximum (μ) transmission angle,respectively.

Therefore, the transmission angle in a four-bar mecha-nism is defined by Equations (34) and (35),

μ = |θ3 − θ4| (34)

if μ >π

2, then μ = π − μ (35)

and the angles θ3 and θ4 are defined by Equations (12) and(17).

It is important to remark that, as the transmission angledecreases, the mechanical advantage is reduced and evena small amount of friction will cause the mechanism tolock. A value close to 90◦ along the cycle of the crank(r2) is acceptable to ensure good quality of the mechanism(Norton 1997). On the other hand, from Figure 5, it can beobserved that the minimum and maximum values for thetransmission angles are defined in Equations (36) and (37).

μmax = arccos

[r2

3 + r24 − (r1 + r2)2

2r3r4

](36)

μmin = arccos

[r2

3 + r24 − (r1 − r2)2

2r3r4

]. (37)

Then, a function is established in Equation (38) in order toevaluate the performance of the mechanism,

�2 = (μmax − 90◦)2 + (μmin − 90◦)2. (38)

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The minimum value of the relationship established inEquation (38) produces best performance of the four-barmechanism.

3.2. Constraints

One of the most important issues to be considered in themechanism design is ensuring that the crank is able tocomplete a cycle. To fulfil the kinematic necessities, a set ofconstraints is proposed. Such set is related to the mechanismsize, symmetry, transmission angle and mobility criterion.

3.2.1. Grashof’s law

Grashof’s law provides that: for a plane four-bar linkage,the sum of the lengths of shortest and largest links cannotbe greater than the sum of the lengths of the two remaininglinks, if a continuous relative rotation between two ele-ments is desired (Norton 1997). Denotingthe shortest andthe largest links of the four-bar mechanism as s and l andthe other two links as p and q, Grashof’s law is establishedas detailed in Equation (39),

s + l ≤ p + q. (39)

In the problem tackled in this paper, Grashof’s law is givenby

r2 + r3 ≤ r1 + r4. (40)

In addition, to ensure that the solution method pro-duces Grashof mechanisms, it must fulfil the conditionsestablished in Equations (41) and (42),

r1 ≤ r3 (41)

r4 ≤ r3. (42)

3.2.2. System size

One of the design considerations used in the four-barmechanism is the maximum size of the links. Due to theavailable space, the length is determined between 0.05 mand 0.5 m, so that the restrictions are presented in Equations(43)–(46),

0.05 ≤ r1 ≤ 0.5 (43)

0.05 ≤ r2 ≤ 0.5 (44)

0.05 ≤ r3 ≤ 0.5 (45)

0.05 ≤ r4 ≤ 0.5. (46)

On the other hand, the angle between the horizontal axisand the reference bar (r1) is limited between 45◦ and −45◦,as pointed out in Equation (47),

− 45◦ ≤ θ1 ≤ 45◦. (47)

3.2.3. Transmission angle

As it was pointed out before, the transmission angle μ mustbe greater than 45◦ along the crank cycle, so that it mustfulfil Equation (48),

45◦ ≤ μ(r1, r2, r3, r4, t), (48)

where the transmission angle is calculated using Equation(34). It is worth remarking that this design constraint isa dynamic constraint. This fact is because the constraintmust be calculated along the whole crank cycle. Thepresence of this type of constraints makes difficult theusage of a mathematical programming method in order toobtain a solution of the optimisation problem unless suchconstraints are static. In Mezura-Montes, Portilla-Flores,Coello Coello, Alvarez Gallegos, and Cruz-Villar (2008),a mathematical programming method was used in orderto solve a mechatronic design problem. From that work,some implementation issues were highlighted regardingconstraints: in a general way, linear approximations of theconstraints were used by the mathematical programmingmethod, this fact kept the approach from generating feasiblesolutions. Furthermore, the mathematical programmingmethod required gradient calculation of the constraints.

3.2.4. Rocker’s displacement symmetry

With an output whose behaviour is symmetric, two advan-tages have to be noticed. The first one consists of an equalmomentum required at each one of the tightening points.The other advantage is presented in an equal displacementthrough both planar movement axes. To ensure this equalityconstraint, Equation (49) is established,

180◦ − θ4max = θ4min. (49)

4. Design of the mechanism

Once the criteria and constraints were established, the setof parameters to apply the kinematic design approach mustbe proposed.

4.1. Design variables

To carry out the parametric four-bar design, the vector �x isdefined in Equation (50),

�x = (x1, x2, x3, x4, x5)T = (r1, r2, r3, r4, θ1)T , (50)

where the first four variables correspond to the lengths of thefour-bar links [m] and the last variable is the angle betweenthe horizontal axis and the reference bar [rad]. With thisset of parameters, the four-bar mechanism is completelydescribed.

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4.2. Optimisation problem

To obtain the mechanical four-bar parameter optimal va-lues, a constrained bi-objective optimisation problem withone dynamic constraint is proposed in Equations (51)–(58),

opt � = [�1(�x),�2(�x)]�x ∈ R5 (51)

with

�1(�x) = −(θ4max − θ4min)2 (52)

�2(�x) =(μmax − π

2

)2+

(μmin − π

2

)2, (53)

subject to

g1(�x) = x2 + x3 − x1 − x4 ≤ 0 (54)

g2(�x) = x1 − x3 ≤ 0 (55)

g3(�x) = x4 − x3 ≤ 0 (56)

g4(�x, t) = π

4− μ(x, t) ≤ 0 (57)

h1(�x) = π − θ4max − θ4min = 0 (58)

0.05 ≤ xi ≤ 0.5, i = 1, . . . , 4, and − π

4≤ x5 ≤ π

4,

(59)

where �1(�x) (the output of the four-bar mechanism) must bemaximised and �2(�x) (the differences of the transmissionangle values with respect to 90◦) must be minimised. �1(�x),the output of the mechanism, must be equal or greater than0. Therefore, this value is squared. Moreover, with the aimto get the two objective functions minimised �1(�x) is mul-tiplied by −1 in Equation (52). Finally, it is important tonotice that the time interval used to evaluate dynamic con-straints and the angles θ3 and θ4 are the time interval of onecrank cycle.

5. Multi-objective modified bacterial foragingoptimisation algorithm (MOMBFOA)

As it was mentioned in Section 1, MBFOA is based onBFOA. Both of them are based on social and cooperativebehaviours bacteria have when looking for regions withhigh nutrient levels (Bremermann 1974). This section startsby explaining the foraging behaviour of bacteria. After that,the original BFOA and its modified version for engineeringdesign problems MBFOA are detailed. Finally, the modifi-cations made to MBFOA to solve the four-bar mechanismdesign problem are presented.

5.1. Bacteria foraging behaviour

Each bacterium tries to maximise its obtained energy pereach unit of time spent on the foraging process while avoid-ing noxious substances. Moreover, bacteria can communi-cate among them. A swarm of bacteria S behaves as follows(Passino 2002):

(1) Bacteria are randomly distributed in the map ofnutrients.

(2) Bacteria move towards high-nutrient regions inthe map by means of chemotaxis (tumble andswimming). Those located in regions with noxioussubstances or low-nutrient regions will die anddisperse, respectively. Bacteria in high-nutrientregions will reproduce (split).

(3) Bacteria are now located in promising regions of themap of nutrients as they try to attract other bacteriaby generating chemical attractors.

(4) Bacteria are now located in the highest nutrientregion.

(5) Bacteria now disperse to look for new nutrientregions in the map.

5.2. Bacterial foraging optimisation algorithm

Four main processes can summarise the bacterial forag-ing behaviour: (1) chemotaxis, (2) swarming, (3) reproduc-tion and (4) elimination-dispersal. Based on them, Passino(2002) proposed BFOA to solve unconstrained numericalsingle-objective optimisation problems. The algorithm isdetailed in Table 2.

A bacterium i represents a potential solution to theoptimisation problem (i.e. a vector of real numbers, whichincludes n design variables of the optimisation problemand identified as �x in Section 4.1) [x1, x2, . . . , xn], and it isdefined as θ i(j, k, l), where j is its chemotactic loop value,k is its reproduction loop value and l is its elimination-dispersal loop value. The tumble within the chemotacticloop of each bacterium i was modeled by Passino with thegeneration of a random search direction φ(i) as presentedin Equation (60),

φ(i) = �(i)√�(i)T �(i)

, (60)

where �(i) is a randomly generated vector of size n withelements within the following interval: [−1, 1]. After that,each bacterium i modifies its position by a swimming stepas indicated in Equation (61),

θ i(j + 1, k, l) = θ i(j, k, l) + C(i)φ(i). (61)

The swim will be repeated Ns times if and only if thenew position is better than the previous one, i.e. (assuming

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Table 2. BFOA pseudocode.

BeginCreate a random initial swarm of bacteria θ i(j, k, l)∀i, i = 1, . . . , Sb

Evaluate f (θ i(j, k, l)) ∀i, i = 1, . . . , Sb

For l=1 to Ned DoFor k=1 to Nre Do

For j=1 to Nc DoFor i=1 to Sb Do

Update f (θ i(j, k, l)) to emulate the swarming processPerform the chemotaxis process(tumble-swim) with Equations (60) and (61)for bacteria θ i(j, k, l) controlled by Ns

End ForEnd ForPerform the reproduction process by sortingall bacteria in the swarm based on f (θ i(j + 1, k, l)),deleting the Sr worst bacteria and duplicating the remaining 1 − Sr

End ForPerform the elimination-dispersal process by eliminatingeach bacteria θ i(j, k, l) ∀i, i = 1, . . . , Nb with probability 0 ≤ Ped ≤ 1

End ForEnd

Note: Input parameters are number of bacteria Sb , chemotactic loop limit Nc , swimloop limit Ns , reproduction loop limit Nre, number of bacteria for reproduction Sr

(usually Sr = Sb/2), elimination-dispersal loop limit Ned, stepsizes Ci (they dependof the dimensionality of the problem) and probability of elimination dispersal ped.

minimisation) f (θ i(j + 1, k, l)) < f (θ i(j, k, l)). Other-wise, a new tumble is computed. The chemotactic loopstops when the chemotactic loop limit Nc has reached for allbacteria in the swarm. The swarming process is based on amodification to the fitness landscape by making more attrac-tive boundaries of the location of a given bacterium with theaim to attract more bacteria to such region. This process re-quires the definition of a set of parameter values by the user(Passino 2002). The reproduction process consists of sort-ing all bacteria in the swarm θ i(j, k, l),∀i, i = 1, . . . , Nb

based on their objective function value f (θ i(j, k, l)) andeliminating half of them with the worst value. The remain-ing half will be duplicated so as to maintain a fixed pop-ulation size. The elimination-dispersal process consists ofeliminating each bacteria θ i(j, k, l),∀i, i = 1, . . . , Nb witha probability 0 ≤ Ped ≤ 1.

5.3. Modified BFOA

MBFOA was proposed with the aim of adapting BFOA forsolving constrained numerical single-objective optimisa-tion problems. Furthermore, the original algorithm was sim-plified and some parameters were eliminated so as to makeit easier to use in engineering design problems (Mezura-Montes and Hernandez-Ocana 2009).

MBFOA adopted a penalty-function-free constraint-handling technique to bias the search to the feasible re-gion of the search space, as well as a generational loop

(G) where three inner processes are carried out: chemo-taxis, reproduction and elimination-dispersal. On the basisof the fact that in MBFOA there are no reproduction andelimination-dispersal loops because they are replaced bythe generational loop, a bacterium i in generation G and inchemotactic step j is defined as θ i(j,G).

The four modifications in MBFOA are detailed as fol-lows:

(1) The constraint handling mechanism modifies theselection criteria used in the tumble-swim opera-tor and in the sorting carried out in the reproduc-tion process. The only criterion used in the originalBFOA was the objective function value. Instead,in MBFOA a set of three rules proposed by Deb(2000) is employed. They are the following:

(a) Between two feasible bacteria, the one withthe best objective function value is pre-ferred.

(b) Between one feasible bacterium and one in-feasible bacterium, the feasible one is pre-ferred.

(c) Between two infeasible bacteria, the onewith the lowest sum of constraint violationis preferred.

The sum of constraint violation is computed as:∑mi=1 max(0, gi(�x)), where m is the number of con-

straints of the problem. Each equality constraint isconverted into an inequality constraint: ‖ hi(x) ‖

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–∈ ≤ 0, where ∈ is the tolerance allowed (a verysmall value).

(2) The chemotactic process consists, as in BFOA, oftumble-swim movements carried out by bacteriain the current swarm. The tumble movement (i.e.search direction chosen at random) is computed inthe same way as in BFOA (see Equation (60)). Theswim movement is similar as that used in BFOA aswell, but based on the modified nomenclature it isnow defined in Equation (62),

θ i(j + 1,G) = θ i(j,G) + C(i)φ(i), (62)

where θ i(j+1,G) is the new position of bacteriumi (new solution) and θ i(j,G) is the current positionof bacterium i.

Unlike BFOA, where the stepsize values in vec-tor C(i) are provided by the user, in MBFOAthey are calculated by considering the valid lim-its per each design variable k (Mezura-Montes andHernandez-Ocana 2009) as indicated in Equation(63),

C(i)k = R∗(

��xk√n

), k = 1, . . . , n, (63)

where ��xk is the difference between upper (Uk)andlower (Lk) limits for design parameter xk: Uk − Lk ,n is the number of design variables and R is a user-defined percentage of the value used by the bacteriaas stepsize.

Furthermore, instead of the swarming process,which requires four user-defined parameters, an at-tractor movement was added to MBFOA so as tolet each bacterium in the swarm to follow the bac-terium located in the most promising region of thesearch space in the current swarm, i.e. the best cur-rent solution. The attractor movement, proposed byMezura-Montes and Hernandez-Ocana (2009) forMBFOA, is presented in Equation (64),

θ i(j + 1,G) = θ i(j,G) + β(θB(G) − θ i(j,G)),(64)

where θ i(j + 1,G) is the new position of bacteriumi, θ i(j,G) is the current position of bacterium i,θB(G) is the current position of the best bacteriumin the swarm so far at generation G and β definesthe closeness of the new position of bacterium iwith respect to the position of the best bacteriumθB(G). The attractor movement applies twice in achemotactic loop, while in the remaining steps thetumble-swim movement is carried out. The aim isto promote a balance between exploration and ex-ploitation in the search. Finally, if the value of adesign variable generated by the tumble-swim or

attractor operators is outside the valid limits de-fined by the optimisation problem, the followingmodification, taken from Kukkonen and Lampinen(2006), is made so as to get the value within thevalid range as shown in Equation (65),

xi =⎧⎨⎩

2∗Li − xi if xi < Li

2∗Ui − xi if xi > Ui

xi otherwise,(65)

where xi is the design variable value i generatedby any bacterium movement and Li and Ui arethe lower and upper limits for design variable i,respectively.

(3) The reproduction process in MBFOA is similar tothat used in BFOA, i.e. the swarm is sorted and thefirst half survives, whereas the second half is elim-inated. The first half is also duplicated to maintaina fixed size of the swarm at each cycle. However, inMBFOA the criteria to sort the swarm of bacteriaare the three rules of the constraint-handling tech-nique and not only the objective function value asin BFOA.

(4) The elimination-dispersal process eliminates onlythe worst bacterium, based on the three criteria de-fined in the constraint-handling technique, and anew randomly generated bacterium is inserted as areplacement.

The complete pseudocode of MBFOA is detailed inTable 3.

5.4. Multi-objective MBFOA

Recalling from Section 4, the problem defined and tack-led in this paper is a constrained bi-objective optimisationproblem with one dynamic constraint. Therefore, MBFOAwas modified so as to get the MOMBFOA. The details arepresented in the following subsections.

5.5. Pareto dominance and constraint-handlingmechanism

It is well documented in the specialised literature that themost suitable criterion to be added to an EA or an SIAso as to solve either a two- or three-objective optimisationproblem is Pareto dominance (Deb 2002), which is definedas follows: a vector of objectives � = [�1, . . . , �K ] domi-nates �′ = [�′

1, . . . , �′K ] (denoted by � � �′) if and only

if � is partially less than �′, i.e. ∀i ∈ {1, . . . , K}, �i ≤�′

i ∧ ∃i ∈ {1, . . . , K} : �i < �′i . The set of all the Pareto

non-dominated solutions is called the Pareto optimal set.The objective function values of those solutions containedin the Pareto optimal set constitute the Pareto front of the

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Table 3. MBFOA pseudocode.

BeginCreate a random initial swarm of bacteria θ i(j, 0)∀i, i = 1, . . . , Sb

Evaluate f (θ i(j, 0)), gm(θ i(j, 0)), ∀i, i = 1, . . . , Sb, ∀m, m = 1, . . . , MFor G=1 to GMAX Do

For i=1 to Sb DoFor j=1 to Nc Do

Perform the chemotaxis process(tumble-swim with Equations (60) and (62)and attractor operator with Equation (64)for bacteria θ i(j, G) by considering the constraint-handling technique

End ForEnd ForPerform the reproduction process by sortingall bacteria in the swarm based on the constraint-handling technique,deleting the Sr worst bacteria and duplicating the remaining 1 − Sr

Perform the elimination-dispersal process by eliminatingthe worst bacterium θw(j, G) in the current swarmby considering the constraint-handling technique

End ForEnd

Note: Input parameters are number of bacteria Sb , chemotactic loop limit Nc , number ofbacteria for reproduction Sr (usually Sr = Sb/2), scaling factor β, percentage of initialstepsize R and number of generations GMAX.

problem. In this paper, as stated in Section 4, the mechanicaldesign problem has two objectives, then K = 2.

The solution of a multi-objective problem can be de-fined as follows: if the feasible region of the search space isnamed asF , either the EA or the SIA will look for the Paretooptimal set: P∗ := {�v ∈ F | ¬∃ �s ∈ F �(�s) � �(�v)}.

In this paper, a real-world problem is being solved, thenP∗ is unknown, a sub-optimal Pareto set, including sub-optimal trade-off solutions for the mechanical design prob-lem is the solution sought.

5.5.1. Selection criteria

As it was found in the specialised literature, constraint han-dling in numerical multi-objective optimisation problemhas received less attention than the single-objective op-timisation problem (Ray, Singh, Isaacs, and Smith 2009;Yen 2009). Nevertheless, there are competitive proposalssuch as the combination of the feasibility rules describedin Section 5.3 with Pareto dominance for non-dominatedsorting (Deb, Pratap, Agarwal, and Meyarivan 2002). Thereare other similar proposals (Oyama, Shimoyama, and Fujii2007), where the third rule considered Pareto dominancein the constraint search space when two solutions are in-feasible. On the basis of its similarity with the constraint-handling technique used in MBFOA, the feasibility rulesdescribed with Pareto dominance (Deb et al. 2002) isadopted in MOMBFOA. This criteria will apply in thetumble-swim movements and in the sorting of the swarm.The set of criteria is defined as follows:

(a) Between two feasible bacteria, the one which dom-inates the other is preferred. If both bacteria arefeasible and non-dominated each other, one is cho-sen at random.

(b) Between one feasible bacterium and one infeasiblebacterium, the feasible one is preferred.

(c) Between two infeasible bacteria, the one with thelowest sum of constraint violation is preferred.

5.5.2. Dynamic constraint handling

One particular aspect of the four-bar mechanism optimaldesign problem introduced in Section 3.2.3 and stated inSection 4 is the presence of a dynamic inequality con-straint (g14(�x)), which keeps the transmission angle μ

over 45◦ along the whole crank cycle. Although the us-age of EAs and SIAs in dynamic optimisation problemshas been documented in the specialised literature (Yang,Ong, and Jin 2007), the treatment of dynamic constraintshas been scarcely investigated. In Kumar-Singh, Isaacs,Thanh-Nguyen, Ray, and Yao (2009), a framework for un-constrained dynamic optimisation was adapted to solveproblems with dynamic constraints. In Nguyen and Yao(2009), constraints were handled by a penalty function anda special operator to convert infeasible solutions into feasi-ble ones. The authors concluded that a special treatment tothe constraints is more suitable than keeping or maintainingdiversity in the population as in unconstrained dynamic op-timisation (Yang et al. 2007). Motivated by the last finding,in this paper, a special treatment is applied to the dynamic

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Table 4. Crowding distance pseudocode.

Input data: archive (A)a=size(A) // store archive sizeFor All (i ∈ A ) do

A[i]dist=0 //initialise crowding distance for all bacteria to 0End ForFor All objective function �j do

A=sort(A,�j ) //Sort A in descending order according to �j

A[1]dist = A[a]dist = ∞ // ∞ value to bacteria in the extremes of the front for objective jFor i=2 to a-1 do // for the remaining bacteria

A[i]dist = A[i]dist+ ‖ �A[i−1]j

−�A[i+1]j

�maxj

−�minj

‖ //compute distance value for objective j

End ForEnd For

constraint as follows: the evaluation of such constraint isdiscretised in such a way it is evaluated along the trajec-tory described by the four-bar mechanism, i.e. one crankcycle. The number of inner evaluations of the dynamic con-straint is 900. After that, the sum of violations for each innerevaluation is computed and added to the sum of constraintviolation for a given bacterium (see Section 5.3).

5.6. External archive

Inspired by the state-of-the-art algorithms for multi-objective optimisation (Zitzler and Thiele 1999; Knowlesand Corne 2000), an external archive to store only feasiblenon-dominated bacteria is considered in MOMBFOA. Thearchive, which is empty at the beginning of the optimisationprocess, is updated at each MOMBFOA cycle by insertinga copy of those feasible bacteria from the swarm after thechemotactic process. Each time a set of bacteria enters thefile, a non-dominance checking is carried out in the archiveto keep only non-dominated bacteria.

5.7. Diversity mechanism

With the aim to maintain a suitable diversity in the swarmof bacteria, as recommended for EAs and SIAs to solvemulti-objective optimisation problems (Deb 2002; CoelloCoello, Van Veldhuizen, and Lamont 2002), and takingcare of not adding additional parameters as considered inthe constraint-handling mechanism integrated with Paretodominance, the crowding distance (Deb 2002) was adoptedand employed in non-dominated feasible solutions in theexternal archive. Crowding distance values are higher forthose more isolated solutions in the objective space. There-fore, bacteria with higher crowding distance values in thearchive will be preferred. Table 4 details the calculation ofsuch distance.

5.8. Attractor operator

Recalling from Section 5.3 and Equation (64), an opera-tor was added to MBFOA to allow the best bacterium inthe swarm to attract other bacteria during their chemotactic

process. In single-objective optimisation, the selection ofthe best bacterium is straightforward because one uniquesolution is sought. However, in multi-objective optimisa-tion, as indicated in Section 5.5, a set of solutions is lookedfor. Therefore, the selection of the best bacterium must bedesigned to bias the search to a sub-optimal Pareto front.Hence, in MOMBFOA the leader will be chosen from theexternal archive and it will be the bacterium with the high-est crowding distance value, i.e. the solution located in themost isolated region of the current Pareto front. The goalis to promote the sampling of more solutions in such re-gion and extend the front. The updated attractor operator isshown in Equation (66),

θ i(j + 1,G) = θ i(j,G) + β(θBCr (G) − θ i(j,G)), (66)

where θ i(j + 1,G) is the new position of the bacterium i,θ i(j,G) is the current position of bacterium i, θBCr (G) isthe current position of the bacterium with the best crowdingdistance in generation G taken from the external file and β

is the stepsize of the attractor movement. If the archive isempty, i.e. no feasible non-dominated solutions have beenfound, the leader is chosen from the current swarm andbased on the criteria defined in Section 5.5.1, i.e. the bac-terium with the lowest sum of constraint violation. The aimthere is to bias the search to the feasible region.

5.9. Reproduction

On the basis of the fact that in MBFOA the reproductionprocess consists of a cloning-like process where an identicalcopy of a bacterium is generated as offspring, the diversityin the swarm might decrease fast. On the other hand, inmulti-objective optimisation diversity must be maintainedbecause a set of solutions (i.e., a Pareto sub-optimal set) issought. Therefore, in MOMBFOA only the best bacteriumin the swarm will reproduce with one copy, which willreplace the second worst bacterium, based on a sorting pro-cess computed by considering the criteria defined in Section5.5.1, i.e. either the bacterium with the second to last highest

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Table 5. MOMBFOA pseudocode.

BeginArchive = ∅Create a random initial swarm of bacteria θ i(j, 0)∀i, i = 1, . . . , Sb

Evaluate �k(θ i(j, 0)),gm(θ i(j, 0)), ∀i, i = 1, . . . , Sb, ∀k, k = 1, . . . , K , ∀m, m = 1, . . . , MFor i=1 to Sb do

If θ i(j, 0) is feasibleAdd θ i(j, 0) to the Archive by using Pareto dominance

End IfEnd ForCompute the crowding distance for all solutions in the ArchiveFor G=1 to GMAX Do

For i=1 to Sb DoFor j=1 to Nc Do

Perform the chemotaxis process(tumble-swim with Equations (60) and (62)and attractor operator with Equation (66) (or 64 if Archive=∅))for bacteria θ i(j, G) by considering the criteria defined in Section 5.5.1

End ForEnd ForPerform the reproduction process by duplicating the bestbacterium in the swarm based on the criteria defined in Section 5.5.1and deleting the second-to-last worst bacteriumPerform the elimination-dispersal process by eliminatingthe worst bacterium θw(j, G) in the current swarmby considering the criteria defined in Section 5.5.1Compute the crowding distance for all solutions in the Archive

End ForEnd

Note: Input parameters are number of bacteria Sb , chemotactic loop limit Nc , scalingfactor β, percentage of initial stepsize R and number of generations GMAX.

sum of constraint violation (if the swarm contains either nofeasible bacteria or a combination of feasible and infeasiblebacteria) or a dominated bacteria (if the swarm has eitheronly feasible bacteria or just one infeasible bacteria). Thecomplete MOMBFOA pseudocode is presented in Table 5.

6. Results and discussion

To evaluate the behaviour and performance of MOMBFOA,four experiments were designed.

(1) Analysis of the constraint satisfaction.(2) Comparison of sub-optimal Pareto fronts.(3) Comparison based on performance metrics.(4) Analysis of the solutions obtained in the sub-

optimal Pareto set per each compared algorithm.

For comparison purposes, four meta-heuristic algo-rithms were chosen to solve the four-bar mechanism op-timal design problem: (1) NSGA-II (Deb et al. 2002), (2)SPEA2 (Zitzler, Laumanns, and Thiele 2002), (3) the DEfor mechanical design (DEMD) (Portilla-Flores et al. 2011)and (4) the multi-objective particle swarm optimisation al-gorithm MOPSO (Reyes-Sierra and Coello Coello 2006).

NSGA-II and SPEA2 were chosen based on the fact thatthey are the most popular EAs to solve multi-objective op-timisation problems. Moreover, NSGA-II uses crowdingdistance as a diversity mechanism, whereas SPEA2 usesan external file to store non-dominated solutions. DEMDwas precisely proposed to solve multi-objective mechan-ical design problems and it also uses crowding distanceas diversity mechanism as well as an external archive forelitism purposes. MOPSO is based on particle swarm op-timisation and has proved to be very competitive to solvemulti-objective optimisation problems in presence of con-straints (Reyes-Sierra and Coello Coello 2006). The fouralgorithms adopted the same constraint-handling techniqueused by MOMBFOA for the dynamic constraint.

Ten independent runs were carried out by each com-pared algorithm (MOMBFOA, NSGA-II, SPEA2, DEMDand MOPSO) in each experiment. The set of parameter val-ues remained fixed for each algorithm in all experimentsand 500,000 evaluations were defined as the terminationcriteria for each independent run.

The parameter values for each algorithm were ob-tained after preliminary experiments by considering dif-ferent combinations and choosing the one that provided thebest performance. A summary of the preliminary tests is

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Table 6. Summary of preliminary results based on the HV metric for parameter selection for each compared algorithm.

Algorithm/parameters Best Mean Median St. Dev. Worst

MOMBFOA. Sb=100, GMAX=250, Nc=20, B=1.7, R=0.0018 0.2233 0.2026 0.2085 0.02419 0.176MOMBFOA. Sb=200, GMAX=125, Nc=20, B=1.2, R=0.0018 0.1836 0.0938 0.0696 0.06 0.0284MOMBFOA. Sb=200, GMAX=100, Nc=25, B=1.7, R=0.0018 0.2803 0.2574 0.2488 0.02 0.2431MOMBFOA. Sb=200, GMAX=250, Nc=10, B=1.7, R=0.0018 0.1591 0.0937 0.1222 0.0832 0MOMBFOA. Sb=250, GMAX=100, Nc=20, B=1.7, R=0.0018 0.2763 0.2699 0.2678 0.0056 0.2657MOMBFOA. Sb=200, GMAX=125, Nc=20, B=1.7, R=0.01 0.2598 0.2373 0.2404 0.02419 0.2117MOMBFOA, Sb=200, GMAX=125, Nc=20, B=1.7, R=0.0025 0.2471 0.2373 0.2439 0.01423 0.221DEMD. NP=200, GMAX=2500 0.2936 0.2796 0.29 0.021 0.2554DEMD. NP=50, GMAX=10000 0.3259 0.3086 0.3127 0.0196 0.2872SPEA-I. Popsize=200, GMAX=2500, C=0.5, M=0.5 0.1356 0.1034 0.0922 0.02827 0.0825SPEA-I. Popsize=200, GMAX=2500, C=1.0, M=0.5 0.1132 0.1066 0.1056 0.006 0.1012SPEA-I. Popsize=200, GMAX=2500, C=0.5, M=1.0 0.0948 0.0881 0.0879 0.0066 0.0816SPEA-I. Popsize=100, GMAX=5000, C=1.0, M=1.0 0.07252 0.0457 0.0562 0.0333 0.00831NSGA-II. Popsize=100, GMAX=5000, C=0.5, M=0.3 0.0974 0.0847 0.0842 0.01245 0.0725NSGA-II. Popsize=100, GMAX=5000, C=0.8, M=0.5 0.1236 0.1089 0.1052 0.0131 0.0981NSGA-II. Popsize=200, GMAX=2500, C=0.8, M=0.3 0.0215 0.0175 0.0192 0.005 0.0118MOPSO. Popsize=200, GMAX=2500, φ1=2.0, φ2=2.0 0.0018 0.0009 0.001 0.0009 0MOPSO. Popsize=100, GMAX=5000, φ1=1.8, φ2=2.5 0.0023 0.0052 0.0017 0.0006 0.0012MOPSO. Popsize=100, GMAX=5000, φ1=1.5, φ2=1.7 0.0102 0.00748 0.0089 0.0036 0.0034MOPSO. Popsize=100, GMAX=5000, φ1=2.5, φ2=1.5 0.0313 0.0216 0.0212 0.0095 0.0123

presented in Table 6, where the hypervolume (HV) metricwas adopted as a criterion to choose the best performanceof each algorithm. This decision was based on the fact thatit does not require the real Pareto front for a single algo-rithm analysis. The preliminary tests were by no meansexhaustive. This was because of the computational timerequired by each single run for each compared algorithmderived mostly from the objective function and constraints(including the dynamic one) evaluations of this real-worldoptimisation problem. The results in Table 6 do not includethose which provided the best results for each algorithm.They are presented in Table 8.

The values for MOMBFOA were Sb = 200, GMAX= 125, Nc = 20 (the tumble-swim operator works in allchemotactic steps for each bacterium with the exception ofsteps 15 and 20, where the attractor operator is used), β =1.7 and R = 1.8E-3. Particulary for MOMBFOA, previousexperiments were carried out to choose the best combina-tion of input parameter values. Different combination ofvalues per parameter were defined and tested in five runsand based on the filtered Pareto fronts obtained the above-mentioned values were adopted in the experiments. It isimportant to remark that the most critical parameter to befine-tuned is R, and part of the future work relies on itsanalysis.

For NSGA-II, the parameter values were popsize =100, GMAX = 5000, crossover rate = 0.8 and muta-tion rate = 0.3. For DEMD, the values were NP = 100,GMAX = 5000, CR ∈ [0.8, 1] generated at random, andF ∈ [0.3, 0.9] also generated at random. The randomnessfor F and CR was adopted because DEMD performs betterwith such approach. For SPEA2, the parameter values were

popsize = 200, GMAX = 2500, crossover rate = 1.0 andmutation rate = 1.0. Finally, for MOPSO, the parametersadopted were popsize = 100, GMAX = 5000, φ1 = 2.5and φ2 = 1.7. The remaining values were kept fixed withsuggested values: repository size = 100, α = 0.1, ngrid =10, β = 4 and γ = 1.

The five algorithms were coded in Matlab R2009b andwere run in a laptop computer with 3GB RAM, Intel CoreDuo processor at 2.8GHz, and Microsoft Windows XP OS.

6.1. Constraint satisfaction

The experimental design related with MOMBFOA’s be-haviour when dealing with the constraints of the problemsconsisted of the following: (1) an analysis of the popula-tion at each generation with respect to the satisfaction ofthe dynamic constraint, (2) an analysis of the population ateach generation with respect to the capability to generatefeasible solutions and (3) the capability to generate feasiblenon-dominated solutions.

The results for the first experiment are summarised inFigure 6, where the average number of solutions in the pop-ulation per generation, which satisfy the dynamic equalityconstraint on 10 independent runs, is plotted.

From Figure 6, three stages can be distinguished inthe behaviour of the approach: (1) between generations 0and 5, where MBFOA quickly gets more than half of itspopulation with solutions satisfying the dynamic constraint,(2) between generations 6 and 40, where another 20% of thepopulation also satisfies the dynamic constraint and finally(3) between generations 41 and 125, where another 15%of the population satisfies the aforementioned constraint

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Figure 6. Average number of solutions per generation on10 independent runs, which satisfy the dynamic constraint.

but with little ‘ups’ and ‘downs’, i.e. slight decrease in thenumber of solutions satisfying the constraint.

The overall behaviour in Figure 6 shows that the pro-posed approach favours the satisfaction of the dynamicconstraint. However, this effect cannot be isolated fromthe capability of the approach to generate feasible solu-tions. Therefore, the average number of feasible solutionsper generation on 10 independent runs are presented inFigure 7, where it is observed that, after generation 22, halfof the population is feasible. After that, between generations23 and 80, another 40% of the population becomes feasi-ble. At the end, between generations 81 and 125, almost theremaining 10% of the population becomes feasible.

Finally, the average number of feasible non-dominatedsolutions stored in the external file per each generation in10 independent runs are plotted in Figure 8.

A similar behaviour with respect to those found in theprevious two experiments is observed, but with a signifi-cant lower number of solutions. Between generations 0 and20, six feasible non-dominated solutions are stored in theexternal file. In the remaining generations, a slower but in-creasing behaviour is observed, where another six feasiblenon-dominated solutions are added to the file.

The overall results of the three experiments regardingthe constraint satisfaction suggest that MBFOA, coupledwith the proposed constraint-handling mechanism designedto also deal with the dynamic constraint of the problem,quickly locates some bacteria of the population in the

Figure 7. Average number of feasible solutions per generationon 10 independent runs.

Figure 8. Average number of feasible non-dominated solutionsper generation in the external file on 10 independent runs.

feasible region of the search space. This behaviour is an at-tractive property of the approach for mechanical engineersbecause a solution can be provided by requiring a low num-ber of evaluations. The way almost the whole populationgradually enters the feasible region is another convenientbehaviour observed in MBFOA because maintaininginfeasible solutions in the population has been reported inthe specialised literature as convenient to avoid local fronts(Ray et al. 2009). This same effect was observed specif-ically for the dynamic inequality constraint, where somesolutions remained infeasible even in the last generations.

6.2. Sub-optimal Pareto fronts comparison

The comparison of the final sub-optimal Pareto fronts ob-tained by each algorithm is presented in Figure 9, wherethe filtered fronts are plotted. Those fronts were obtainedby applying non-dominance checking to all the solutionsfrom the 10 fronts obtained in the 10 independent runs peralgorithm.

From the graphical analysis, it is clear that both MOMB-FOA and DEMD outperformed NSGA-II, MOPSO andSPEA2. MOMBFOA found some feasible solutions in thecentral and left regions in its front, which dominate so-lutions located in the central part of the DEMD’s front.On the other hand, DEMD was able to generate moresolutions in the upper left region of the objective space,whereas MOMBFOA reached the lower right region. The

Figure 9. Filtered Pareto sub-optimal fronts obtained from10 independent runs for MOMBFOA, DEMD, NSGA-II, MOPSOand SPEA2.

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results of this experiment suggest better solutions providedby MOMBFOA. NSGA-II generated three well-defined re-gions of a disconnected sub-optimal Pareto front, whereasSPEA2 generated a widely spaced front with the lowestvalue for the second objective function. It is important toremark that those few solutions obtained by MOPSO areinfeasible (constraints g2 and h1 are violated).

6.3. Comparison based on performance metrics

Performance metrics in nature-inspired multi-objective op-timisation are a valuable tool to analysing the performanceof algorithms (Zitzler, Thiele, Laumanns, Fonseca, and daFonseca 2003). In this paper, three metrics will be employedto analyse the results found by each compared algorithm andthey are described in the following.

• Two set coverage (C-metric): This binary perfor-mance measure estimates the coverage proportion,in terms of percentage of dominated solutions, be-tween two sub-optimal Pareto sets. Given two sets A

and B, both containing only feasible non-dominatedsolutions, the C-metric is formally defined as

C(A,B) = |{u ∈ B|∃v ∈ A : v � u}||B| . (67)

The C-measure value means the portion of solutionsin B being dominated by any solution in A.

• Hypervolume: Having a sub-optimal Pareto setPFknown, and a reference point in objective spacezref, this performance measure estimates the hyper-volume attained by it. The HV corresponds to thenon-overlapping volume of all the hypercubes formedby the reference point (zref) and every vector in thePareto set approximation. This is formally definedas

HV = {∪ivoli |veci ∈ PFknown}, (68)

where veci is a non-dominated vector from the sub-optimal Pareto set, and voli is the volume for thehypercube formed by the reference point and the non-dominated vector veci . A high value for this measureindicates that the sub-optimal Pareto set is closer tothe true Pareto front and that it covers a wider exten-sion of it.

• Generational distance (GD): It is a way of estimat-ing how far are the elements of the sub-optimal Paretofront obtained by one algorithm with respect to thoseof the true Pareto front of the problem. It is definedas follows:

GD =√∑N

i=1 d2i

N, (69)

where N is the number of non-dominated solutionsin the sub-optimal Pareto front obtained by the algo-rithm and di is the Euclidean distance between eachpoint in such front and its nearest member of the truePareto front. Due to the fact that a real-world problemis being solved in this paper, the true Pareto front willbe approximated by applying non-dominance check-ing to the combined set of the accumulated Paretofronts obtained in a set of independent runs by eachcompared algorithm. A value of GD = 0 points outthat all the elements of the sub-optimal front gener-ated by an algorithm are in the true Pareto front. Asthe GD value increases, so is the distance between thesub-optimal and true Pareto fronts. Therefore, lowervalues indicate a better performance of an algorithm.

The statistical results obtained for the C-metric are sum-marised in Table 7. Such results are based on all the pair-wise combinations of the 10 independent runs executedby each algorithm (each one of the 10 sub-optimal Paretofronts of one algorithm was compared with each one the10 fronts obtained by the other algorithm). Mann–Whitneynon-parametric test was used to get confidence on the dif-ferences observed in the samples of runs (Derrac, Garcıa,Molina, and Herrera 2011).

The results in Table 7 show that MOMBFOA andDEMD outperform NSGA-II, MOPSO and SPEA2 in thefour-bar mechanism optimisation problem. It is worth not-ing that NSGA-II and SPEA2 showed a similar performancebased on the C-metric and that MOPSO was competitiveonly with respect to NSGA-II. In the same way, such metricis unable to determine the superiority between MOMBFOAand DEMD.

The results for the HV metric on 10 independent runsare summarised in Table 8. The reference point to calcu-late that metric was (0,0.7). On the basis of the fact that theMann–Whitey test gives 95% confidence on the differencesobserved in such table, DEMD outperformed the other fouralgorithms, whereas MOMBFOA outperformed NSGA-II,SPEA2 and MOPSO. Those findings confirm the observa-tions made in Figure 9, where DEMD generated a largersub-optimal Pareto front with more solutions with respectto that generated by MOMBFOA.

To further analyse the behaviour of the four algorithmswith respect to this metric, Figure 10 shows the averageHV value from two independent runs at different evaluationnumbers. The values confirm the superiority of DEMD withrespect to the HV metric.

Table 9 summarises the statistical values on 10 indepen-dent runs by each compared algorithm for the GD metric.The Mann–Whitney test indicates that there is no signif-icant difference between the results of MOMBFOA andDEMD. The same applies for the results between NSGA-IIand SPEA2. In the remaining pairwise combinations, thedifferences are significant; it means that the MOMBFOA

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Table 7. Summary of results for pairwise comparisons among the four algorithms based on the C-metric.

Mann–WhitneyAlgorithm Best Mean Median St. Dev. Worst test

C(MOMBFOA,NSGA-II) 1 1 1 0 1C(NSGA-II-MOMBFOA) 0 0 0 0 0

√C(DEMD-NSGA-II) 1 1 1 0 1C(NSGA-II-DEMD) 0 0 0 0 0

√C(MOMBFOA-DEMD) 0.9259 0.3366 0.3537 0.2571 0C(DEMD-MOMBFOA) 1 0.3283 0.25 0.3266 0 =C(SPEA2-DEMD) 0 0 0 0 0C(DEMD-SPEA2) 1 0.3413 0.1111 0.4101 0

√C(SPEA2-NSGA-II) 1 0.1 0 0.1005 0C(NSGA-II-SPEA2) 1 0.1917 0 0.3263 0 =C(SPEA2-MOMBFOA) 0.125 0.0033 0 0.0193 0C(MOMBFOA-SPEA2) 1 0.3072 0.0909 0.4035 0

√C(MOMBFOA-MOPSO) 0.5455 0.4191 0.4978 0.1792 0.2141C(MOPSO-MOMBFOA) 0 0 0 0 0

√C(DEMD-MOPSO) 0.5455 0.4237 0.4912 0.1661 0.2345C(MOPSO-DEMD) 0 0 0 0 0

√C(SPEA2-MOPSO) 0.5455 0.3256 0.4312 0.2877 0C(MOPSO-SPEA2) 0.0714 0.0238 0 0.0412 0

√C(NSGA-II-MOPSO) 0 0 0 0 0C(MOPSO-NSGA-II) 0.5455 0.3898 0.3895 0.1555 0.2345

√

Note: ‘√

’ indicates a significant difference based on the Mann–Whitney test with 95% confidence. ‘=’ means no significant difference based on theaforementioned test.

Table 8. Summary of results for the five algorithms based on the HV metric.

Algorithm Best Mean Median St. Dev. Worst Mann–Whitney test

DEMD 0.3548 0.3052 0.3013 0.0266 0.2731√

NSGA-II 0.1478 0.1070 0.1077 0.0311 0.0478√

SPEA2 0.2252 0.1068 0.1718 0.0896 0√

MOPSO 0.0778 0.02593 0 0.0449 0√

MOMBFOA 0.2890 0.2698 0.2750 0.0164 0.2465√

Note: ‘√

’ indicates that the differences observed in the statistical values reported in this table are significant, based on the Mann–Whitney test with 95%confidence, with respect to the other four algorithms (best results are remarked in boldface).

and DEMD provide a similar better performance with re-spect to those observed by NSGA-II, SPEA2 and MOPSO.

6.4. Analysis of solutions

With the aim to provide a more detailed review of the solu-tions provided by MOMBFOA and the four other compared

Figure 10. Average HV value versus number of evaluations pereach compared algorithm.

algorithms (DEMD, NSGA-II, SPEA2 and MOPSO) someof them, located in the region of the objective space withmore solutions generated by the compared algorithms, wereanalysed and subject to simulation so as to get more infor-mation on their suitability in the solution of the four-barmechanism. The solutions chosen were those located at themiddle of each front within the region with more solutions.Figure 11 remarks in a rectangle such region in the objectivespace (�1 ∈ [0.4, 0.8] and �1 ∈ [0.3, 0.5]).

Table 10 encloses those five representative solutionsby each compared algorithm. Those solutions, except withthat infeasible one provided by MOPSO, were introducedin a simulator of the four-bar mechanism. The inputs for thesimulator are the five design variables defined in Section 4.1(x1, x2, x3, x4, and x5).

From a mechanical point of view, angle θ4 must havea displacement over the vertical axis, i.e. by consideringa reference angle of 90◦ (1.5707 rad) θ4 must have a pos-itive/negative value variation. In the same way, the trans-mission angle μ must apply a strong impulse force to the

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Figure 11. Region where more feasible non-dominated solutionswere generated by the compared algorithms.

bar where angle θ4 is located, i.e. the μ angle must oscillateby starting with a high value (close to 1 rad), decreasingto a similar value (1 rad) and be closer with respect to theθ4 angle value. As a consequence, the movement of themechanism is improved.

Regarding the value taken by the displacement angle θ4,its variation must be between −45◦ and 45◦, i.e. between135◦ (2.3561 rad) and 45◦ (0.7854 rad) with respect to thereference angle value of 90◦ (1.5707 rad). On the otherhand, the transmission angle μ, for a good performance ofthe four-bar mechanism, must have a minimum value of 45◦.

The simulation results for the displacement angle θ4

and the transmission angle μ, based on the representativesolutions in Table 10, are shown in Figure 12, where thelower and upper horizontal dashed lines represent the lower45◦ (0.7854 rad) and upper 135◦ (2.3561 rad) limits allowedfor θ4 and μ angles, and the horizontal dotted line in themiddle of the graph indicates the reference value of 90◦

(1.5707 rad).On the basis of the information in Figure 12, four out of

five algorithms obtained valid solutions, i.e. the lower andupper limits for both angles are not violated. However, thesolution provided by SPEA2 is at the edge of the lower limitvalue for the μ angle (0.7854 rad). Regarding closenessbetween angles, the solutions by MOMBFOA and DEMDwere the best ones, whereas the solutions by NSGA-II andSPEA2 show an increasing distance after 0.3 sec.

The solutions obtained by MOMBFOA and DEMD startand end the μ angle value with 1 rad. Meanwhile, the solu-tions reached by NSGA-II and MOMBFOA start and end

Table 10. Details of representative solutions by each comparedalgorithm.

Algorithm x1/m x2/m x3/m x4/m x5/rad

DEMD 0.4439 0.0505 0.4464 0.1406 –0.1295NSGA-II 0.3213 0.0745 0.3220 0.2054 –0.3013SPEA2 0.4041 0.1246 0.386 0.3033 –0.4164MOPSO** 0.4741 0.1107 0.4704 0.2947 –0.3234MOMBFOA 0.4825 0.0631 0.4825 0.1543 –0.1467

∗∗ MOPSO solution is slightly infeasible.

with lower values for the μ angle. Finally, between the so-lution by MOMBFOA and the one obtained by DEMD, thefirst one has the higher amplitude for the displacement ofthe rocker θ4, i.e. it reaches a value of 2.0 rad. Further-more, the transmission angle μ remains more time over 90◦

(the ideal behaviour). These observations confirm the com-ments made on Figure 9, where even with a lower numberof solutions, MOMBFOA was able to find some of themthat dominate those found by DEMD.

7. Conclusions and future work

The synthesis of a four-bar mechanism based on a kine-matic analysis and the optimisation of its design by usingthe multi-objective modified bacterial foraging optimisa-tion algorithm were presented. The design problem wasdefined by taking into account two objective functions andstructural constraints, where at least one of them was adynamic constraint.

MOMBFOA adopted Pareto dominance and the feasi-bility rules as selection criteria to bias the approach to thefeasible region of the search space. Furthermore, the dy-namic constraint was specially treated so as to promote itssatisfaction by evaluating it 900 times so as to get a goodapproximation of its violation. An external file was usedto store feasible non-dominated solutions found during thesearch and crowding distance was used for diversity main-tenance. Finally, the reproduction of bacteria was limited toa very small number of clones per generation and an attrac-tor operator was added to the chemotactic cycle so as to letbacteria to explore the neighbourhood of isolated feasiblenon-dominated solutions.

Table 9. Summary of results for the four algorithms based on the Generational Distance metric.

Algorithm Best Mean Median St. Dev. Worst Mann–Whitney test

DEMD 0.000850 0.007123 0.006300 0.005317 0.014500√

(except with MOMBFOA)NSGA-II 0.120000 0.035200 0.017150 0.054988 0.191000

√(except with SPEA2)

SPEA2 0.039000 0.103078 0.068600 0.086284 0.313000√

(except with NSGA-II)MOPSO 0.029000 1.098743 0.0488 1.903072 3.296222

√MOMBFOA 0.002200 0.010120 0.010200 0.004933 0.021100

√(except with DEMD)

Note: ‘√

’ indicates that the differences observed in the statistical values reported in this table are significant, based on the Mann–Whitney test with 95%confidence, with respect to the other four algorithms (best results are remarked in boldface).

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Figure 12. Simulator output of the angular displacement of the rocker θ4 and transmission angle μ for each feasible representativesolution in Table 10.

Four experiments were carried out to test the behaviourand performance of MOMBFOA in the solution of the real-world mechanical design problem. MOMBFOA was ableto generate feasible solutions (including the dynamic con-straint) early in the optimisation process, but at the sametime infeasible solutions were maintained and this balancecaused, as it was later observed, better final results. Theresults obtained by MOMBFOA were compared with re-spect to those obtained by three state-of-the-art algorithmsfor multi-objective optimisation (NSGA-II, SPEA2 andMOPSO) and one EA designed to solve multi-objective me-chanical design problems (DEMD). The visual comparisonof the accumulated sub-optimal Pareto fronts obtained byeach algorithm showed that the generated by MOMBFOA,even with a lower number of non-dominated solutions, wasbetter with respect to those generated by DEMD, NSGA-II, SPEA2 and MOPSO. On the other hand, the comparisonbased on three performance metrics (C-metric, HV and GD)showed that MOMBFOA and DEMD were clearly betterthan NSGA-II, SPEA2 and MOPSO. Moreover, MOMB-FOA and DEMD were equally good based on the C-metricand GD, whereas DEMD was slightly better than MOMB-FOA for the HV metric due to its larger sub-optimal Paretofront. Finally, a review of representative solutions locatedin the most populated region of the objective space showedthat the solution obtained by MOMBFOA presented the bestperformance in a simulation-based analysis, where both an-gles related with the input force generated by the four-barmechanism had the most competitive values.

From a theoretical point of view, the complexity ofMOMBFOA is similar to that of SPEA2, O(G(N + A)2)

(Jensen 2003), where G is the number of generations of thealgorithm, N is the population size and A is the archive size.The similarities lie in the usage and update of the externalarchive where feasible non-dominated solutions are stored.DEMD also has a similar complexity due to its archiveusage, whereas NSGA-II has a complexity of O(GMN2)(Jensen 2003).

The results presented in this paper validated the modelproposed for the four-bar mechanism. Furthermore, theoverall behaviour of MOMBFOA (smaller sub-optimalPareto front but with better solutions) showed to be suitablefor this type of mechanical design problems.

Part of the future work relies on the incorporation ofthe design engineer preferences into MOMBFOA. More-over, MOMBFOA will be used to solve other real-worldmechanical design problems, specially those related withconcurrent design, where both, the mechanical design andthe control of the mechanism, are optimised at the sametime. Finally, MOMBFOA will be analysed to determine thesensitivity to its parameter values, mainly with respect tothat related to the stepsize used in the chemotactic loop (R).

AcknowledgementsThe first author acknowledges support from the Mexican Coun-cil for Science and Technology (CONACyT) through project no.79809. The second author acknowledges support from the COFAAand EDI programmes of the Instituto Politecnico Nacional throughproject no. SIP-20111027 and support from SNI-CONACyT. Thethird author acknowledges support from CONACyT through ascholarship to pursue graduate studies at LANIA.

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Notes on contributorsEfren Mezura-Montes is a full-time re-searcher at the National Laboratory forAdvanced Informatics (LANIA) A.C. inXalapa, Veracruz, Mexico. His researchinterests include the design, analysis andapplication of nature-inspired algorithmsto solve complex optimisation problems.He has published over 70 papers in peer-reviewed journals and conferences. He also

has one edited book published and nine book chapters publishedby international publishing companies.

Edgar A. Portilla-Flores is a full-timeresearcher at the Centro de Innovacion yDesarrollo Tecnologico en Computo delInstituto Politecnico Nacional (CIDETEC-IPN) in Mexico City, Mexico. His researchinterests include integrated design ofmechatronic systems, design of roboticgrippers and mechanical systems of powertransmission.

Betania Hernandez-Ocana graduated withhonours with an MsC on Applied Comput-ing at LANIA A.C. and is currently a PhDstudent at the Universidad Juarez Autonomade Tabasco, Mexico. Her research interestsinclude nature-inspired algorithms for opti-misation.

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Optimum synthesis of a four-bar mechanism using the modified bacterial foraging algorithm

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