ORIGINAL ARTICLE

A novel particle swarm optimization approach for productdesign and manufacturing

Ali Rıza Yıldız

Received: 30 May 2007 /Accepted: 25 February 2008 / Published online: 18 April 2008# Springer-Verlag London Limited 2008

Abstract This paper presents a novel optimization ap-proach that is a new hybrid optimization approach based onthe particle swarm optimization algorithm and receptorediting property of immune system. The aim of the presentresearch is to develop a new optimization approach andthen to apply it in the solution of optimization problems inboth the design and manufacturing areas. A single-objectivetest problem, tension spring problem, pressure vesseldesign optimization problem taken from the literature andtwo case studies for multi-pass turning operations aresolved by the proposed new hybrid approach to evaluateperformance of the approach. The results obtained by theproposed approach for the case studies are compared with ahybrid genetic algorithm, scatter search algorithm, geneticalgorithm, and integration of simulated annealing andHooke-Jeeves pattern search.

Keywords Particle swarm algorithm . Receptor editing .

Hybrid approach .Multi-pass turning . Design optimization

1 Introduction

In today’s manufacturing world, it is a vital task to defineoptimal machining parameters for achieving machining costand efficiency. In order to meet today’s global marketrequirements and to produce higher-quality products at

lower cost with shorter lead times, there is a need tointroduce new optimization concepts. An important prob-lem is how to achieve lower machining cost for machiningeconomics problem consists in determining the processparameters of the metal cutting, usually cutting speed, feedrate, and depth of cut, in order to optimize objectivefunction subject to machining constraints.

In the initial study on the machining economics problems,Gilbert [1] presented theoretical analysis of the optimizationof the machining process. Although most of the earlierstudies were limited to single-pass operations [2–4] due tothe economic reasons, in practice, multi-pass operations arepreferred [5]. Several optimization techniques such asdynamic programming [6–8], geometric programming [9–10], linear programming [11], sequential unconstrainedminimization technique [12], linear programming, andbranch-and-bound [13] have been used for multi-passoperations in the literature. These optimization techniquesare either stuck at local optimum or take a long time toconverge to a reasonable result [14]. The above-mentionedinefficiency of traditional methods in optimization ofcomplex engineering problems has forced researchers tosearch for new approaches.

The convergence speed of evolutionary algorithms to theglobal (or nearly global) optimal results is better than thatof traditional techniques. Therefore, evolutionary algo-rithms like genetic algorithms (GA), differential evolutionalgorithm, ant colony algorithm, immune algorithm andparticle swarm optimization algorithm (PSO) have beenused to improve further the solution of optimizationproblems with complex nature in many applications ofboth metal cutting and design optimization [15–41].

In this regard, Chen and Tsai [5] used integration ofsimulated annealing and Hooke-Jeeves search method foroptimization of operating conditions in multi-pass turning.

Int J Adv Manuf Technol (2009) 40:617–628DOI 10.1007/s00170-008-1453-1

A. R. Yıldız (*)Mechanical Engineering Department, Uludag University,Bursa 16059, Turkeye-mail: aliriza@uludag.edu.tr

Present address:A. R. YıldızMechanical Engineering Department, University of Michigan,Ann Arbor, MI 48109, USA

Onwubolu and Kumalo [27] used the GA to optimize multi-pass turning problem based on the mathematical model ofChen and Tsai [5]. Vijayakumar et al. [28] solved the sameproblem using ant colony optimization technique. Chen andChen [29] revealed that Onwubolu and Kumalo [27] did notconsider the equality constraint based on total depth of cut.Similarly, Vijayakumar et al. [28] did not consider theequality constraint on total depth of cut. Since Onwuboluand Kumalo [27] and Vijayakumar et al. [28] did notconsider the equality constraint based on total depth of cut,their results are not compared with those of Chen and Tsai[5]. Chen [30] applied scatter search algorithm to theoptimization of machining economics models in multi-passturning operations. A hybrid genetic algorithm (HRGA) wasdeveloped to optimize machining parameters for multi-passturning operations by Yildiz and Ozturk [18].

The PSO algorithm has been used in many areas ofoptimization studies [19–20, 23, 26, 31–42]. It was appliedto shape optimization of torque arm and to the sizeoptimization of truss structures taken from literature byFourie and Groenwold [20]. In their PSO algorithm, theconcept of craziness was redefined and elitism operatorborrowed by GA was used. Their results showed that PSOalgorithms were better than GA and the gradient-basedrecursive quadratic programming algorithm.

Tandon et al. [19] used the PSO algorithm to optimizemachining parameters in end milling. Onwubolu and Clerc[23] used PSO algorithms to minimize the operation path inthe computer controlled drilling operations. Karpat andÖzel [26] have shown an approach using PSO to obtainmachining data by optimizing the surface roughness andtool life in turning. Haq et al. [31] used the PSO algorithmto optimize the machining allocation of clutch assembly.Their results indicated that PSO algorithm was better thangeometric programming (GP) and genetic algorithm (GA)in terms of decreasing the manufacturing cost for twodimensional optimal machining tolerance allocation ofclutch assembly. The PSO algorithm is used to model endmilling process and to predict the surface roughness in endmilling by El-Mounayri et al. [32].

Although the evolutionary algorithms offer significantadvantages over the traditional techniques, they may havepremature convergence towards a local minimum. In orderto overcome the weaknesses of evolutionary algorithms andto avoid premature convergence towards a local minimum,evolutionary optimization techniques have been hybridizedand commonly used in the optimization of manufacturingand design optimization problems [18, 21, 24, 41–45].Yildiz et al. [21] were hybridized GA with Taguchi methodto solve multi-objective shape optimization problems.

It is known that the PSO algorithm is more efficient thanGA at exploring the solution space, but it does notguarantee the global optimum as other evolutionary

methods. The introduction of hybrid methods comes fromthe need to tackle more and more complex real-worldproblems. Some of the hybrid approaches in literature havebeen made on hybrid particle swarm [41–42].

Fan et al. [41] proposed a hybrid approach algorithmbased on the Nelder-Mead simplex search method and thePSO algorithm. The approach was applied to the optimizationof multimodal functions. Their results indicated that theproposed algorithm was better than hybrid genetic algorithm(GA), continuous GA, simulated annealing, and tabu search infinding optimal solutions for multimodal functions.

Xia andWu [42] proposed a hybrid approach based on thehybridization of the PSO and SA and they applied to themulti-objective flexible job-shop scheduling problem as acase study.

Although some improvements regarding optimization ofcutting parameters in multi-pass turning operations havebeen achieved, due to the complexity of machine parame-ters with conflicting objective and constraints, optimizationof the machining economic problems still presents animportant topic of investigation. Therefore, in recent years,there has been a growing interest in the new hybridapproaches to optimize machining economic problems.

The aim of this research is to further develop and strengthenthe PSO algorithm and to apply it to both the optimization ofcutting parameters for multi-pass turning operations anddesign optimization problems. In this research, a new hybridapproach based on PSO algorithm and receptor-editingproperty of immune system is presented. The efficiency ofthe proposed hybrid approach is evaluated by single-objectivetest problem, tension spring, pressure vessel design optimiza-tion problems taken from the optimization literature.

Finally, the proposed hybrid approach is applied to thetwo case studies to optimize cutting parameters and tominimize the unit production cost in multi-pass turningoperations. In addition, in order to show that the proposedoptimization approach can be effectively applied to real-world optimization problem from both manufacturing anddesign industry, also an application of the proposedapproach to shape optimization problems from automotiveindustry is given in the Appendix.

The organization of the paper is as follows. Section 3presents a detailed formulation of the objective andconstraints in multi-pass turning. The standard PSO, immunesystem and receptor editing are presented in Sect. 4. Section 5proposes the hybrid PSO approach. The proposed PSO hasbeen tested on tree test and design problems that arecommonly used in the optimization literature in Sect. 6.Two case studies are solved. The results and discussions forcase studies are given in Sect. 7. The paper is concluded inSect. 8. An application of the proposed hybrid approach toshape optimization of an automobile component fromautomotive industry is given in the Appendix.

618 Int J Adv Manuf Technol (2009) 40:617–628

2 Nomenclature

The notation used in the machining model is defined asfollows:

C0 Constant pertaining to tool-life equation,CI Machine idle cost ($/piece),CM Cutting cost by actual time in cut ($/piece),CR Tool replacement cost ($/piece),CT Tool cost ($/piece),dr, ds Depths of cut for each pass of rough and finish

machining (mm),drL, drU Lower and upper bounds of depth of rough cut

(mm),dsL, dsU Lower and upper bounds of depth of finish cut

(mm),dt Total depth of metal to be removed (mm),D Diameter of workpiece (mm),fr, fs Feeds in rough and finish machining (mm/rev),frL, frU Lower and upper bounds of feed in rough

machining (mm/rev),fsL, fsU Lower and upper bounds of feed in finish

machining (mm/rev),Fr, Fs Cutting forces during rough and finish

machining (kg f),FU Maximum allowable cutting force (kg f),h1, h2 Constants pertaining to tool travel and approach/

depart time (min),k1, k2,k3

Constants for roughing and finishing parameterrelations,

kf Coefficient pertaining to specific tool-workpiececombination,

ko Direct labor cost overhead ($/min),kq Coefficient pertaining to equation of chip-tool

interface temperature,kt Cutting edge cost ($/edge),L Length of workpiece (mm),n Number of rough passes,p, q, r Constants pertaining to the tool-life equation,Pr, Ps Cutting power during roughing and finishing

(kW),PU Maximum allowable cutting power (kW),Qr, Qs Temperatures during roughing and finishing (0C),QU Maximum allowable temperature (0C),Ra Maximum allowable surface roughness (mm),Rn Nose radius of cutting tool (mm),Sc Limit of stable cutting region,t Tool life (min),tc Constant term of machine idling time (min),te Tool exchange time (min),tp Tool life (min) considering roughing and

finishing,

tr, ts Tool lives (min) for roughing and finishing,tv Variable term of machine idling time (min),TI Machine idling time (min),TL, TU Lower and upper bounds of tool life,TM Cutting time by actual machining (min),TMr,TMs

Cutting time by actual machining for roughingand finishing (min),

TR Tool replacement time (min),UC Unit production cost except material cost ($/

piece),Vr, Vs Cutting speeds in rough and finish machining

(m/min),VrL ;VrU Lower and upper bounds of cutting speed in

rough machining (m/min),VsL ;VsU Lower and upper bounds of cutting speed in

finish machining (m/min),X Vector of machining parameters,τ,φ,δ Constants pertaining to expression of chip-tool

interface temperature,η Power efficiency,1,ν Constants pertaining to expression of stable

cutting region,μ,υ Constants of cutting force equation.

3 Optimization model of multi-pass turning operations

3.1 The objective function

In multi-pass turning operations, the aim is to minimize unitproduction cost (UC). The unit production cost is the sumof the cutting cost (CM), machine idle cost (CI), toolreplacement cost (CR) and tool cost (CT). The mathematicalmodel of the case study for multi-pass turning operationswas defined by Chen and Tsai [5]. More details about themathematical model can be obtained from Shin and Joo [8]and Chen and Tsai [5]. The unit production cost (UC) isdefined as follows:

UC ¼ CM þ CI þ CR þ CT ð1Þ

UC ¼ k0πDL

1000Vrfr

dt � dsdr

� �þ πDL

1000Vsfs

� �

þk0 tc þ ðh1Lþ h2Þ dt � dsdr

þ 1

� �� �

þk0teTp

πDL1000Vrfr

dt � dsdr

� �þ πDL

1000Vsfs

� �

þ ktTp

πDL1000Vrf r

dt � dsdr

� �þ πDL

1000Vsfs

� �

ð2Þ

Int J Adv Manuf Technol (2009) 40:617–628 619

3.1.1 Rough machining

Bounds on depth of cut : drL � dr �drU ð3Þ

Bounds of feed : f rL � f r � f rU ð4Þ

Bounds of cutting speed : VrL � Vr � VrU ð5Þ

Tool� life constraint : TL � tr �TU ð6Þ

Cutting force constraint : Fr ¼ klfμr d

υr � FU ð7Þ

Power constraint : Pr ¼ k1fμr d

υr Vr

6120η� PU ð8Þ

Stable cutting region constraint : V1r f rd

νr � SC ð9Þ

Chip� tool interface temperature constraint :

Qr ¼ kqVtRf

φr d

δr � QU

ð10Þ

3.1.2 Finish machining

Bounds on depth of cut : dsL � ds � dsU ð11Þ

Bounds of feed : f sL � fs � f sU ð12Þ

Bounds on cutting speed : VsL � Vs � VsU ð13Þ

Tool� life constraint : TL � ts � TU ð14Þ

Cutting force constraint : Fs ¼ k1fμs d

υs � FU ð15Þ

Power constraint : Ps ¼ k1fμs d

υsVS

6120η� PU ð16Þ

Stable cutting region constraint : V1s f sd

νs � SC ð17Þ

Chip� tool interface temperature constraint :

QS ¼ k2VCs f

φs d

δs � QU

ð18Þ

Surface finish constraint :f 2s8R

� SRU ð19Þ

3.2 Parameter relations

Vs � k3Vr ð20Þ

f r � k4f s ð21Þ

dr � k5ds ð22Þ

where k3, k4, k5 are relationship coefficients and k3, k4,k5 ≥ 1.

dr ¼ dt � dsð Þ=n ð23Þ

In addition to these constraints, the total depth of cut isanother important constraint for the case study. The totaldepth of cut (dt) is the sum of the depth of finish cut (ds)and the total depth of rough cut (ndr). The optimizationalgorithm does not determine the optimal depth of roughingsince it can be given by the mathematical manipulation asexpressed in Eq. (24).

ds ¼ dt � ndr ð24Þ

Therefore, the equality constraint and the decisionvariable (dr) and (n) in the optimization procedure can beeliminated. The five machining parameters (Vr, fr, ds, Vs,fs) are determined for turning model optimization. Furtherdetails about the turning mathematical model and data withrespect to machining can be obtained from Shin and Joo [8]and Chen and Tsai [5] and Chen [30].

4 Particle swarm optimization algorithm and immunesystem

4.1 Particle swarm optimization algorithm

The particle swarm optimization (PSO) algorithm, whichwas inspired by the social behavior of bird flocking or fishschooling, is a population-based optimization techniquedeveloped by Eberhart and Kennedy [46]. In the PSOalgorithm, each individual or solution, called a "particle",has a velocity and position in problem space search and allof the particles together are called a "swarm". The positionsof the particles change according to the velocity given inEq. (25). The velocity is updated by the particle’s own

620 Int J Adv Manuf Technol (2009) 40:617–628

experience and the experience of the particle’s neighbors orthe experience of the swarm.

Vi;kþ1 ¼ W � Vi;k þ c1r1 Pi;k � Xi;k

� �þ c2r2 Gk � Xi;k

� � ð25ÞThe velocity equation is the sum of the old velocity, the

own experience of the particle, and the interaction withgroup, respectively. Vi,k+1 is the updated velocity forparticle i, representing the distance to be traveled by thisparticle from its current position. Xi,k represents the particleposition. Two variants used in the PSO algorithm are globalneighborhood and local neighborhood. The global neigh-borhood Gk, called the global best solution (gbest), is thebest particle position in the whole swarm. On the otherhand, local neighborhood Pi,k, called the local best solution(lbest), is the own best position of each particle [46].Although the number of gbest is equal to one, the numbersof lbest’s are the same as that of particle in the each swarm.

W is called the "inertial factor" and regulates theexchange between the global exploration and local exploi-tation abilities of the swarm. It is typically chosen in therange of [0,1]. c1 and c2, called the social and cognitiveparameters, are used to adjust the second and third terms invelocity equality. They usually use r1 and r2, twoindependent numbers, which are chosen from 0 to 1randomly.

The PSO algorithm includes three steps, namely,generating particles’ positions and velocities, velocityupdate, and position update.

Xi;kþ1 ¼ Xi;k þ Vi;k ð26ÞAfter the velocity of the each particle is updated, the

positions of the particles are updated according to Eq. (26).The initial position and velocity vectors for the particles aregenerated according to Eqs. (27) and (28), respectively.

Xi;k ¼ Xmin þ Xmax � Xminð Þ � r1 ð27Þ

Vi;k ¼ Vmin þ Vmax � Vminð Þ � r2 ð28ÞThe optimization loop continues until termination con-

dition is satisfied. The values assigned to each of theparameters in the proposed hybrid approach are tabulated inTable 1.

For further details about PSO algorithm can be foundfrom [46].

4.2 The immune system

The natural immune system has a vital role in keeping ourliving body healthy from infectious foreign organismscalled antigens such as viruses, bacteria, and other para-sites. The immune system needs to distinguish all cells (ormolecules) within our body and categorize those cells asour own or foreign. While the disease-causing external cellsare named as "nonself", its own harmless cells are called"self". There are several types of immune cells in theimmune system. Lymphocytes are the main type of immunecells and are classified as “T” and “B” cells. B-cells arecapable of recognizing antigens free in solution (e.g., in theblood stream), whereas T-cells can recognize via otheraccessory cells [47].

4.2.1 Receptor editing

Receptor editing mechanism introduces diversity, helps toescape from local optima on an affinity landscape, andleads to possible new candidates. In Fig. 1, a simplifiedlandscape is illustrated.

In Fig. 1 [48], the x-axis shows all possible antigen-binding sites, with the most similar adjacent to each other.The y-axis shows the antigen-antibody affinity. When it istaken a particular antibody (A) selected during the primaryimmune response, point mutations permit the immunesystem to explore local regions around A by making smallsteps towards an antibody with higher affinity. As a result,the antibodies might become stuck at a local optimum (A’)[47–50].

Table 1 PSRE parameter values

Parameter Value

W 1.4C1 0.6C2 1.7

Fig. 1 Schematic representation of shape-space for antigen-bindingsites [48]

Int J Adv Manuf Technol (2009) 40:617–628 621

Receptor editing allows an antibody to take large stepsthrough the landscape. Although this will land the antibodyin a local optimum (B), occasionally the leap will lead to anantibody on the side of a higher hill (C), where the climbingregion have more chance to reach to the global optimum.From this point (C), the antibodies can reach to the top ofthe hill (C’). In conclusion, receptor editing may rescueimmune responses stuck on unsatisfactory local optima([47–50].

In receptor editing mechanism, a percentage of theantibodies (refer to particles in PSO) in the antibodypopulation (refer to swarm in PSO) are replaced withrandomly created antibodies. This mechanism allowsfinding new particles that corresponds to new searchregions in the all search space. Exploring new searchregions may help the algorithm to escape from localoptimal.

5 The proposed hybrid approach

In this paper, a new hybrid optimization approach (ParticleSwarm and Receptor Editing-PSRE) is presented. Theproposed approach incorporates particle swarm algorithmand receptor editing property of immune system.

Although the PSO algorithm is very effective at findingrelatively good neighborhoods of solution in a complexsearch space, they may have premature convergencetowards a local minimum [41, 42]. In this research, inorder to prevent the premature convergence of the PSOalgorithm and to increase diversity, it is hybridized withreceptor editing property of artificial immune system.

In the proposed new hybrid approach, after the positionsof the particles are updated, receptor editing is applied tothe swarm to escape from local optimal. In the receptorediting, new particles are produced and they are replacedwith the particles chosen randomly from the swarm. If theparticle chosen from the swarm is gbest, this particle is notreplaced with new particle. Instead of gbest, another particleis chosen from the swarm.

The algorithm of the proposed approach can be outlinedas follows:

BEGIN

Step 1: Define problemStep 2: Particle swarm algorithm

2.a Generate initial swarm population with randompositions and velocities

While maximum iterations or minimum error criteria isnot attained

for i:=1 to NOP (number of particle) do begin2.b Calculate fitness value for each particle2.c For each particle,If the fitness value is better than its the best fitness value

(lbest) in history set current value as the new lbest2.d Determine the best fitness value of all the particles

as the gbest2.e Calculate velocity of every particle according to

Eq. (25)2.f For each particleUpdate position of every particle according to Eq. (26)

end;

Step 3: Receptor editingfor i:=1 to (0.35* number of the particles) do begin

3.a Apply receptor editing to randomly selected particlesend;

EndEND.

6 Evaluation of proposed approach usingengineering optimization problems

In order to evaluate the performance of the proposed hybridapproach, single-objective test problem, tension springproblem, pressure vessel design optimization problemscommonly used in the literature of engineering optimizationare successfully solved by the proposed algorithm. Theresults of the benchmark problems are compared with those

Table 2 Statistical results of different methods for the first test problem

Methods Best Mean Worst Std Dev Function Evaluations

PSRE −15 −14.876 −14.6819 0.113 100,000Coello and Cortes [44] −14.7841 −14.5266 −13.8417 0.2335 150,000Yoo and Hajela [43] −5.2735 −3.7435 −2.4255 0.9696 150,000Hamida and Schoenauer [51] −15 −14.84 N.A N.A 1,500,000Koziel and Michalewicz [52] −14.7864 −14.7082 −14.6154 N.A 1,000,000Hadj-Alouane and Bean [53] −5.16559 −3.64004 −2.72518 0.60624 N.AMichalewicz and Attia [54] −7.34334 −5.07136 −3.59536 0.77247 N.A

622 Int J Adv Manuf Technol (2009) 40:617–628

of other methods that are representative of the state-of-the-art in the optimization literature. After it is shown that theproposed approach is successful to optimize the complexengineering optimization problems, it is applied to two casestudies for multi-pass turning operations.

6.1 Single-objective test problem

The first test problem is a minimization problem for asingle-objective function with 13 variables and nineinequality constraints. This problem is employed by severalauthors to evaluate the performances of their approaches[43–44, 55–58] and it is defined as follows:

Minimize:

f xð Þ ¼ 5X4i¼1

xi � 5X4i¼1

x2i � 5X13i¼5

xi

Subject to:

g1ðxÞ ¼ 2x1 þ 2x2 þ x10 þ x11 � 10 � 0g2ðxÞ ¼ 2x1 þ 2x3 þ x10 þ x12 � 10 � 0g3ðxÞ ¼ 2x2 þ 2x3 þ x11 þ x12 � 10 � 0g4ðxÞ ¼ �8x1 þ x10 � 0g5ðxÞ ¼ �8x2 þ x11 � 0g6ðxÞ ¼ �8x3 þ x12 � 0g7ðxÞ ¼ �2x4 � x5 þ x10 � 0g8ðxÞ ¼ �2x6 � x7 þ x11 � 0g9ðxÞ ¼ �2x8 � x9 þ x12 � 0

where 0≤xi≤1 (i = 1,...,9), 0≤xi≤100 (i = 10,11,12) and0≤x13≤1. The global optimum is at x*=(1,1,1,1,1,1,1,1,1,3,3,3,1) where f(x*)=−15

The parameters used by the proposed hybrid approachfor optimization process are the following:

(a) number of particles: 100;(b) maximum number of generations: 1,000;(c) number of objective function evaluations: 100,000.

The efficiency of each approach can be measured bycomparing the number of function evaluations, which isequal to the number of particles (population size) multipliedby the number of generations as each solution is evaluatedonce in every generation. In the evolutionary optimizationtechniques, the aim is to find the optimum results withminimum number of function evaluations. In this paper, thePSRE is run 30 times independently.

The results of the proposed hybrid approach (PSRE) forfirst test problem are compared against those provided byCoello and Cortes [44]. PSRE gives better solutions for thisproblem compared with those given in Table 2 as far as thenumber of function evaluations, the best solution computed,and the statistical analysis results are taken into accounttogether. The statistical values of PSRE show the robust-ness of the solutions with the standard deviation of 0.113.The standard deviation is very low, which indicates thatPSRE is among the most robust approach for finding anoptimum solution.

The PSRE requires only 100,000 function evaluations tofind the best-known solution of −15. Coello and Cortes [44]found the best solution −14.7841 for 150,000 functionevaluations with standard deviation of 0.2335. In addition,Coello and Cortes [44] solved the same problem with Yooand Hajela’s [43] approach to compare with their approach.Coello and Cortes [44] found the best solution −5.2735

P D

d

P

Fig. 2 Design variables for ten-sion spring problem [57]

Table 3 Comparison of the best solution for the spring design problem by different methods

Design variables PSRE Coello and Montes [58] Coello [57] Arora [56] Belungdu [55]

x1(d) 0.0516904023 0.051989 0.051480 0.053396 0.050000x2(D) 0.3567500 0.363965 0.351661 0.399180 0.315900x3(p) 11.2871200 10.890522 11.632201 9.185400 14.25000g1(x) −0.0000058368 −0.000013 −0.002080 0.000019 −0.000014g2(x) −0.0000010045 −0.000021 −0.000110 −0.000018 −0.003782g3(x) −4.0537944322 −4.061338 −4.026318 −4.123832 −3.938302g4(x) −0.7277134115 −0.722698 −4.026318 −0.698283 −0.756067f(x) 0.01266527 0.0126810 0.0127048 0.127303 0.0128334

Int J Adv Manuf Technol (2009) 40:617–628 623

with standard deviation of 0.9696 with Yoo and Hajela’s[43] approach.

The use of the PSRE improves the convergence rate bycomputing the best value −15 with respect to very lowstandard deviation 0.113 and maintaining the less functionevaluations 100,000.

6.2 Tension spring optimization problem

This problem was described by Belegundu [55] andconsists of minimizing the weight of a tension springsubject to constraints on minimum deflection, shear stress,surge frequency, limits on outside diameter and on designvariables. The design variables are the mean coil diameter(x1), the wire diameter (x2) and the number of active coils(x3) as shown in Fig. 2.

This problem is studied by several authors to evaluatethe performance of their approaches [55–58]. The mathe-matical formulation of this problem can be described asfollows:

Minimize:

f ðxÞ ¼ ðx3 þ 2Þx2x21 ð25ÞSubject to:

g1ðxÞ ¼ 1� x32x371785x41

� 0 ð26Þ

g2ðxÞ ¼ 4x22 � x1x212566ðx2x31 � x41Þ

þ 1

5108x21� 1 � 0 ð27Þ

g3ðxÞ ¼ 1� 140:45x1x22x3

� 0 ð28Þ

g4ðxÞ ¼ x2 þ x11:5

� 1 � 0 ð29Þ

An exterior penalty approach [55], a numerical optimi-zation technique called constraint correction at constantcost [56], a GA-based coevolution model [57] and afeasibility-based tournament selection scheme [58] wereapplied to this problem. The parameters for optimizationsearch by PSRE are the following:

(a) number of particles: 60;(b) maximum number of generations: 500;(c) number of objective function evaluations: 30,000.

PSRE provides better solutions for this problem com-pared with those given in Tables 3 and 4 and improves theconvergence rate by computing the best value 0.01266527with respect to very low standard deviation 6.24 E-6 andmaintaining the less function evaluations 30,000.

Table 4 Statistical results of different methods for the spring design problem

Methods Best Mean Worst Std Dev Function Evaluations

PSRE 0.01266527 0.012673 0.012708 6.24 E-6 30,000Coello and Montes[58] 0.0126810 0.0127420 0.012973 5.9 e-5 80,000Coello [57] 0.0127048 0.012769 0.012822 3.939 e-5 900,000Arora [56] 0.0127303 N/A N/A N/A N/ABelungdu [55] 0.0128334 N/A N/A N/A N/A

Th Ts

L

R R

Fig. 3 Design variables forpressure vessel [57]

624 Int J Adv Manuf Technol (2009) 40:617–628

6.3 Pressure vessel optimization problem

This problem was proposed by Kannan and Kramer [59]. Acylindrical vessel is capped at both ends by hemisphericalheads as shown in Fig. 3.

The objective is to minimize the total cost, including thecost of the material, forming and welding. There are fourdesign variables: Ts (thickness of the shell), Th (thickness ofthe head), R (inner radius), and L (length of the cylindricalsection of the vessel, not including the head). Ts and Th areinteger multiples of 0.0625 in., which are the availablethicknesses of rolled steel plates, and R and L are continuousvariables. In the mathematical model of pressure vesseldesign, Ts, Th, R and L parameters are represented x1, x2, x3and x4, respectively. This problem is studied by severalauthors to evaluate the performance of their approaches [57–61]. The mathematical model of pressure vessel optimizationproblem can be described as follows:

Minimize:

f ðxÞ ¼ 0:6224x1x3x4 þ 1:7781x2x23 þ 3:1661x21x4

þ 19:84x21x3 ð30ÞSubject to:

g1ðxÞ ¼ �x1 þ 0:0193x3 � 0 ð31Þ

g2ðxÞ ¼ �x2 þ 0:00954x3 � 0 ð32Þ

g3ðxÞ ¼ �px23x4 �4

3px33 þ 1296000 � 0 ð33Þ

g4ðxÞ ¼ x4 � 240 � 0 ð34Þ

A branch-and-bound technique [60], an augmentedLagrangian multiplier approach [59], a genetic adaptivesearch [61], a GA-based coevolution model [58], and afeasibility-based tournament selection scheme [57] wereapplied to this problem. The parameters for optimizationprocess are the following:

(a) number of particles: 60;(b) maximum number of generations: 500;(c) number of objective function evaluations: 30,000.

PSRE provides better solutions for this problem as givenin Tables 5 and 6. The worst solution found by PSRE isbetter than the best solutions found by Kannan and Kramer[59] and found by Sandgren [60]. The PSRE improves theconvergence rate by computing the best value 6059.7144and maintaining the less function evaluations 30,000.

Table 5 Comparison of the best solution for pressure vessel design problem by different methods

Design variables PSRE Coello and Montes [58] Coello [57] Deb [61] Kannan and Kramer [59] Sandgren [60]

x1(Ts) 0.8125 0.812500 0.812500 0.937500 1.12500 1.125000x2(Th) 0.4375 0.437500 0.437500 0.500000 0.62500 0.625000x3(R) 42.09844559 42.097398 40.323900 48.329000 58.29100 47.70000x4(L) 176.6366 176.654050 200.00000 112.679000 43.69000 117.7010g1(x) −0.00000019 −0.000020 −0.034324 −0.004750 0.000016 −0.20439g2(x) −0.03588092 −0.035891 −0.052847 −0.038941 −0.06890 −0.169942g3(x) −0.03590969 −27.886075 −27.10584 −3652.8768 −21.2201 54.22601g4(x) −63.3632788 −63.345953 −40.00000 −127.32100 −196.3100 −122.299f(x) 6059.7144 6059.9463 6288.7445 6410.3811 7198.0428 8129.1036

Table 6 Statistical results of different methods for pressure vessel design problem

Methods Best Mean Worst Std Dev Function evaluations

PSRE 6059.7144 6097.446 6156.57 35,781 30,000Coello and Cortes [34] 6061.1229 6734.0848 7368.0602 457.9959 150,000Coello and Montes [58] 6059.9463 6177.2533 6469.3220 130.9297 80,000Coello[57] 6288.7445 6293.8432 6308.1497 7.4133 900,000Deb [61] 6410.3811 N/A N/A N/A N/AKannan and Kramer [59] 7198.0428 N/A N/A N/A N/ASandgren [60] 8129.1036 N/A N/A N/A N/A

Int J Adv Manuf Technol (2009) 40:617–628 625

7 Case study of computational machining optimization

The proposed hybrid approach is applied and evaluated forsolving the turning optimization problem with the modeland data provided in the references of Shin and Joo [8],Chen and Tsai [5] and Chen [30]. The two case studies forthe multi-pass turning operation, described in Sect. 3, areoptimized using the proposed hybrid approach. Theconstant values and parameters for case 1 are presented inTable 7. The total dept of cutting for case 1 is set with dt=6and it is set with dt=8 for case 2. The parameters used bythe PSRE for optimization search process are the following:

(a) number of particles: 50;(b) maximum number of generations: 200;(c) number of objective function evaluations: 10,000.

From the comparison of best results given in Table 8, itis seen that the minimization of the unit production cost inmulti-pass turning operation is achieved by proposedhybrid approach. The comparison of the results obtainedby the proposed approach, against other techniques, isgiven in Table 8. It can be seen that better results for thebest computed solutions are achieved for the turningoptimization problem compared to HRGA (hybrid geneticalgorithm), SS (scatter search), FEGA (float encoding GA)and SA/PS (Simulated Annealing and Hooke-Jeeves pattern

search) as shown in Table 8 [5, 29, 30]. The results of thisresearch indicate that PSRE can offer a helpful potential asan alternative approach for applications to the optimizationof design and manufacturing problems.

8 Conclusions

The aim of this research is to develop a new optimizationapproach to solve optimization problems in the areas ofdesign and manufacturing by hybridizing the particle swarmoptimization algorithm with receptor editing property ofimmune system. The new hybrid approach (PSRE) issuccessfully applied to the optimization of both designoptimization problem and machining parameters consideringminimum production cost under a set of machining con-straints in multi-pass turning operation. The results ofstatistical analysis and best-known values of both benchmarkand case studies indicate that the PSRE is a good alternativein solving of real-world optimization problems finding bettersolutions compared to other approaches that are represen-tative of the state-of-the-art in the optimization literature.The PSRE is a generalized optimization method. Therefore,it can also be applied to optimization of the other metalcutting problems such as milling, drilling, grinding etc., anddesign and shape optimization problems.

Table 7 Data for the exampleof multi-pass turning Multi-pass turning data

D=50 mm L=300 mm dt=6.0 mmVrU=500 m/min VrL=50 m/min frU=0.9 mm/revfrL=0.1 mm/rev drU=3.0 mm drL=1.0 mmVsU=500 m/min VsL=50 m/min fsU=0.9 mm/revfsL=0.1 mm/rev dsU=3.0 mm dsL=1.0 mmko=0.5 $/min kt=2.5 $/edge h1=7×10

−4

h2=0.3 tc=0.75 min/piece te=1.5 min/edgep=5 q=1.75 r=0.75C0=6×10

11 TU=45 min TL=25 minkf=108 μ=0.75 υ=0.95η=0.85 FU=200 kg f PU=5 kWλ=2 ν=−1 Sc=140kq=132 τ=0.4 φ=0.2δ=0.105 QU=1,000°C Rn=1.2 mmRa=10µ k1=1.0 k2=2.5k3=1.0

Table 8 Comparison of the best computed optimum results for turning problem (PSRE: particle swarm and receptor editing; HRGA: hybridgenetic algorithm; SS: scatter search; SA/PS: simulated annealing and Hooke-Jeeves pattern search; FEGA:float encoding genetic algorithm)

Methods PSRE HRGA [21] SS [30] SA/PS [5] FEGA [29]

Case 1: cost ($) (dt=6 mm) 2.0470 2.0481 2.0667 2.2795 2.2988Case 2: cost ($) (dt=8 mm) 2.4793 2.4086 2.5417 2.7411 2.8170

626 Int J Adv Manuf Technol (2009) 40:617–628

Appendix

An example from the automobile industry

The example is considered to evaluate the PSRE and toillustrate how it can be applied for shape optimization ofautomobile components. Most of the shape optimizationproblems in automotive industry are usually multi-objective,often conflicting. The aim of the problem is to find optimumshape of the side door part of a commercial vehicle. Furtherdetails about figure and design variables of the side door partcan be found in [21]. The objective functions are due to thevolume and the frequency of the part that is to be designedfor minimum volume and avoiding critical frequencysubject to strength constraints. Multi-objective shapeoptimization is performed using PSRE. The comparison ofthe results provided by PSRE, against other techniques, isgiven in Table 9.

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Table 9 Comparison of the design optimization results for side doorbracket

Methods X1

(mm)

X2

(mm)

X3

(mm)

X4

(mm)

Volume

(cm3)

Frequency

(Hz)

Stress

(MPa)

Proposed

hybrid

approach

(PSRE)

30 26.78 13.47 45.1 16525 9.32 294

Hybrid

genetic

algorithm

[21]

30 26.7 12.6 44.4 16754 9.3 291

Genetic

algorithm

(GA)

26.8 26.5 11.6 42.3 17663 8.8 293

CAD

optimum

design

22 27 13.1 36.7 18763 8.1 277

Topology

design

18 24 11 32 20727 8.3 256

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