PLEASE SCROLL DOWN FOR ARTICLE

Transcription

1 This article was downloaded by: [Yuan Ze University] On: 22 October 2008 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Engineering Optimization Publication details, including instructions for authors and subscription information: Solving constrained optimization problems with hybrid particle swarm optimization Erwie Zahara a ; Chia-Hsin Hu a a Department of Industrial Engineering and Management, St. John's University, Tamsui, Taiwan, Republic of China Online Publication Date: 01 November 2008 To cite this Article Zahara, Erwie and Hu, Chia-Hsin(2008)'Solving constrained optimization problems with hybrid particle swarm optimization',engineering Optimization,40:11, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Engineering Optimization Vol. 40, No. 11, November 2008, Solving constrained optimization problems with hybrid particle swarm optimization Erwie Zahara* and Chia-Hsin Hu Department of Industrial Engineering and Management, St. John s University, Tamsui, Taiwan 251, Republic of China (Received 17 August 2007; final version received 9 January 2008 ) Constrained optimization problems (COPs) are very important in that they frequently appear in the real world. A COP, in which both the function and constraints may be nonlinear, consists of the optimization of a function subject to constraints. Constraint handling is one of the major concerns when solving COPs with particle swarm optimization (PSO) combined with the Nelder Mead simplex search method (NM- PSO). This article proposes embedded constraint handling methods, which include the gradient repair method and constraint fitness priority-based ranking method, as a special operator in NM-PSO for dealing with constraints. Experiments using 13 benchmark problems are explained and the NM-PSO results are compared with the best known solutions reported in the literature. Comparison with three different metaheuristics demonstrates that NM-PSO with the embedded constraint operator is extremely effective and efficient at locating optimal solutions. Keywords: constrained optimization; Nelder Mead simplex search method; particle swarm optimization; constraint handling 1. Introduction Constrained optimization is a mathematical programming technique that attempts to solve nonlinear objective functions, or deals with constraints that have a nonlinear relationship, or handles both at the same time. It has become an important branch of operations research and has a wide variety of applications in such areas as the military, economics, engineering optimization and management science (Nash and Sofer 1996). Constrained optimization problems (COPs) with n variables and m constraints may be written in the following canonical form: Minimize f(x) (1) subject to g j (x) 0,j = 1,...,q h j (x) = 0,j = q + 1,...,m l i x i u i,i = 1,...,n *Corresponding author. erwi@mail.sju.edu.tw ISSN X print/issn online 2008 Taylor & Francis DOI: /

3 1032 E. Zahara and C.-H. Hu where x = (x 1,x 2,,x n ) is a one-dimensional vector of n decision variables, f(x) is an objective function, g j (x) 0 are q inequality constraints and h j (x) = 0 are m q equality constraints. The values u i and l i are the upper and the lower bounds of x i, respectively. Traditionally, most of the methods found in the literature search for approximate solutions to COPs and assume that the goal and constraints are differentiable (Fung et al. 2002). In the case that the goal and constraints are not differentiable, heuristic methods are employed. Optimization methods using heuristics, which have helped researchers formulate such new stochastic algorithms as genetic algorithms (Tang et al. 1998), evolutionary algorithms (Runarsson and Yao 2005) and particle swarm optimization (PSO) (Dong et al. 2005), are of little practical use in solving COPs due to slow convergence rates. Fan and Zahara (2007) have demonstrated the hybrid Nelder Mead simplex method and particle swarm optimization method (NM-PSO) to be a promising and viable tool for solving unconstrained nonlinear optimization problems. This article demonstrates that this algorithm can also be adopted to solve constrained nonlinear optimization problems. Until recently, the most commonly used constraint handling technique for stochastic optimization was the penalty method (Coello Coello 2002). The penalty method converts a COP to an unconstrained problem by adding, to the objective function, the constraints as a penalty term. When a solution is deemed infeasible, meaning that the solution satisfies the constraints partially, the objective value is penalized by a penalty set beforehand. The difficulty of using this method is in the selection of an appropriate penalty value for the problem at hand. To overcome this drawback, some researchers have suggested replacing the constant penalty by a variable penalty sequence according to the degree of constraint violation (Farmani and Wright 2003). However, this type of scheme usually ends up with the difficulty of selecting another set of parameters (Coello Coello 2002). In contrast, a constraint fitness priority-based ranking method (Dong et al. 2005) attempts to deal with the difficulty of selecting an appropriate penalty value by constructing a constraint fitness function. A constraint fitness priority-based ranking method (Dong et al. 2005) and a repair method using the gradient information derived from the constraint set (Chootinan and Chen 2006) are embedded in NM-PSO to evaluate the particles during the search when solving COPs. The superior performance of the proposed NM-PSO will be demonstrated by solving 13 benchmark COPs. The remainder of this article is organized as follows. The gradient repair method and constraint fitness priority-based ranking method for handling constraints are briefly discussed in Section 2. The framework of hybrid Nelder Mead simplex search and PSO are detailed in Section 3. In Section 4, the results obtained in this study are compared against the optimal or best known solutions of the 13 benchmark problems reported in the literature. The application of the proposed method to a ceramic grinding optimization problem is discussed in Section 5; finally, conclusions are drawn in Section 6 based on the results. 2. Constraint handling methods This section introduces two constraint handling methods: the gradient repair method and constraint fitness priority-based ranking method, which will be embedded in NM-PSO The gradient repair method The gradient repair method was proposed by Chootinan and Chen (2006). It utilizes gradient information derived from the constraint set to systematically repair infeasible solutions by directing infeasible solutions toward the feasible region. The procedure is summarized as follows.

4 Engineering Optimization 1033 (1) For any solution, determine the degree of constraint violation V by Equation (2). [ ] [ ] g V = q 1 Min{0, g(x)} = V = (2) h (m g) 1 h(x) m 1 where V consists of vectors of inequality constraints (g) and equality constraints (h) for the problem. (2) Compute x V, where x V are the derivatives of these constraints with respect to the solution vector (n decision variables) and e is a small scalar for perturbation. [ ] [ ] x g x V = 1 g(x xi x h e = x i + e) g(x), i = 1, 2,...n (3) h(x x i = x i + e) h(x), i = 1, 2,...n m n Therefore, the relationship between changes of constraint violation V and solution vector x is expressed by m n V = x V x = x = x V 1 V (4) (3) Compute the Moore Pensore inverse or pseudoinverse ( x V +), which is the approximate inverse of x V to be used instead in Equation (3). (4) Update the solution vector by x t+1 = x t + x = x t + x V 1 V x t + x V + V. (5) Loop through steps 1 to 4 at most five times. The above procedure is demonstrated using the following example in which there are four inequality constraints and one equality constraint, given by Equation (5). g 1 : x1 2 + x x1 2 g 2 : x x x 1 2x 2 x 1 3 g 3 : x 1 0 = V = x 1 g 4 : x 2 0 x 2 = 1 0 xv = 1 0 (5) 0 1 h 1 : x 1 + x 2 5 = 0 x 1 + x (1) Suppose the first iteration produces [x 1,x 2 ]=[6, 0], which clearly violates g 1, g 2 and h 1. Calculate the degree of constraint violation V V = 6 = V = (2) Compute x V and the Moore Pensore inverse or pseudoinverse ( x V +) for the violated constraints [ ] x V = 1 0 = x V = (3) Update the solution vector by x t+1 = x t + x x t + x V + V. [ [ ] 11 x 2 = x x = + 3 = 0] [ ]

5 1034 E. Zahara and C.-H. Hu (4) The same process is repeated at most five times until the solution is finally moved to [x 1,x 2 ]= [3, 2], which is in the feasible region Constraint fitness priority-based ranking method This method, proposed by Dong et al. (2005), solves the difficulty of selecting an appropriate penalty value that appears in the penalty method. The method introduces a constraint fitness function Cf (x) for constraints, and it is computed from both inequality and equality constraints as follows. For inequality constraints g j (x) 0, 1, If g j (x) 0 C j (x) = 1 g j (x) g max (x), If g (6) j (x) >0 where g max (x) = max{g j (x), j = 1, 2, q} and C j (x) is the fitness level of point x for constrained condition (j). For equality constraints h j (x) = 0, 1, If h j (x) = 0 C j (x) = 1 h j (x) h max (x), If h (7) j (x) = 0 where h max (x) = max { h j (x), j = q + 1,q + 2,...m }. The constraint fitness function evaluated at point x is equal to the weighted sum of the C j s, as defined below: m m Cf (x) = w j C j (x), w j = 1, 0 w j 1/m, j. (8) j=1 j=1 where w j is a randomly generated weight for constraint j. The sum signifies the fitness level of point x as related to the feasible domain Q; that is, if Cf (x) = 1, it is an indication that x Q, and on the contrary, if 0 <Cf(x) <1, the smaller Cf (x) is, the less likely that x is in the feasible domain Q or the further away x is from Q. 3. Hybrid Nelder Mead and particle swarm optimization The goal of integrating the Nelder Mead simplex method (NM) and particle swarm optimization (PSO) is to combine their advantages and avoid disadvantages. For example, the NM is a very efficient local search procedure but its convergence is extremely sensitive to the selected starting point; PSO belongs to the class of global search procedures but requires much computational effort (Fan and Zahara 2007) The Nelder Mead simplex search method The NM method, proposed by Nelder and Mead (1965), is a local search method designed for unconstrained optimization without using gradient information. The operations of this

6 Engineering Optimization 1035 method rescale the simplex based on the local behaviour of the function by using four basic procedures: reflection, expansion, contraction and shrinkage. Through these procedures, the simplex can successively improve itself and zero in on the optimum. The steps of NM, which are modified to handle a minimization COP, are as follows. (1) Initialization. An initial N + 1 vertex points are randomly generated in their respective search range or space. Evaluate the fitness of the objective function and the constraint fitness at each vertex point of the simplex. For the maximization case, it is convenient to transform the problem into the minimization case by pre-multiplying the objective function by 1. (2) Reflection. Determine X high and X low, vertices with the highest and the lowest function values, respectively. Let f high,f low and Cf high,cf low represent the corresponding objective function values and constraint function values, respectively. Find x cent, the centre of the simplex without x high in the minimization case. Generate a new vertex x refl by reflecting the worst point according to x refl = (1 + α)x cent αx high (α > 0) (9) Nelder and Mead suggested that α = 1. If Cf refl < 1 and Cf refl >Cf low or Cf refl = 1 and f refl <f low, then proceed with expansion; otherwise proceed with contraction. (3) Expansion. After reflection, two possible expansion cases need to be considered. (3.1) If Cf refl < 1 and Cf refl >Cf low, the simplex is expanded in order to extend the search space in the same direction and the expansion point is calculated by Equation (10) x exp = γ x refl + (1 γ)x cent (γ > 1) (10) Nelder and Mead suggested γ = 2. If Cf exp >Cf low or Cf exp = 1, the expansion is accepted by replacing x high with x exp ; otherwise, x refl replaces x high. (3.2) If Cf refl = 1 and f refl <f low, the expansion point is calculated by Equation (10). If Cf exp = 1 and f exp <f low, the expansion is accepted by replacing x high with x exp ; otherwise, x refl replaces x high. (3.3) Exit the algorithm if the stopping criteria are satisfied; otherwise go to step (2). (4) Contraction. After reflection, there are two possible contraction cases to consider. (4.1) If Cf refl < 1 and Cf refl Cf low, the contraction vertex is calculated by Equation (11): x cont = βx high + (1 β)x cent (0 <β<1) (11) Nelder and Mead suggested β = 0.5. If Cf cont >Cf low or Cf cont = 1, the contraction is accepted by replacing x high with x cont ; otherwise do shrinking in step (5.1). (4.2) If Cf refl = 1 and f refl f low and if f refl f high, then x refl replaces x high and contraction by Equation (11) is carried out. If f refl >f high, then direct contraction without the replacement of x high by x refl is performed. If Cf cont = 1&f cont <f low, the contraction is accepted by replacing x high with x cont ; otherwise do shrinking in step (5.2). (4.3) Exit the algorithm if the stopping criteria are satisfied; otherwise go to step (2). (5) Shrinkage. (5.1) Following step (4.1). in which Cf cont Cf low and contraction has failed, try shrinkage attempts for all points except x low by Equation (12): x i δx i + (1 δ)x low (0 <δ<1) (12) Nelder and Mead suggested δ = 0.5. (5.2) Contraction has failed in step (4.2) where Cf cont < 1orCf cont = 1 but f cont f low. Shrink the entire simplex except x low by Equation (12). (5.3) Exit the algorithm if the stopping criteria are satisfied; otherwise go to step (2).

7 1036 E. Zahara and C.-H. Hu 3.2. Particle swarm optimization Particle swarm optimization (PSO) is one of the latest evolutionary optimization techniques developed by Eberhart and Kennedy (1995). The PSO concept is based on a metaphor of social interaction such as bird flocking or fish schooling. Similar to genetic algorithms, PSO is also population-based and evolutionary in nature, with one major difference from genetic algorithms in that it does not implement filtering, i.e. all members of the population survive through the entire search process. PSO simulates a commonly observed social behaviour, where members of a group tend to follow the leader of the best among the group. The procedure of PSO is reviewed below. (1) Initialization. Randomly generate a swarm of potential solutions called particles and assign a random velocity to each. (2) Velocity update. The particles are then flown through hyperspace by updating their own velocity. The velocity update of a particle is dynamically adjusted, subject to its own past flight and those of its companions. The particle s velocity and position are updated using V New id x New id (t + 1) = c 0 Vid old (t) + c 1 rand() (p id (t) xid old (t)) + c 2 rand() (p gd (t) xid old (t)) (13) (t + 1) = xid old New (t) + Vid (t + 1) (14) where c 1 and c 2 are two positive constants; c 0 is an inertia weight and rand( ) is a random value inside (0, 1). Eberhart and Shi (2001) and Hu and Eberhart (2001) suggested c 1 = c 2 = 2 and c 0 =[0.5 + (rand()/2.0)]. Equation (13) illustrates the calculation of a new velocity for each individual. The velocity of each particle is updated according to its previous velocity (V id ), the particle s previous best location (p id ) and the global best location (p gd ). Particle velocities on each dimension are clamped to a maximum velocity V max, which is a fraction in the search space on each dimension. Equation (14) shows how each particle s position is updated in the search space Hybrid NM-PSO method Embedding the constraint handling methods in NM-PSO is now presented. The population size of this hybrid NM-PSO approach is set at 21N + 1 when solving an N-dimensional problem. The initial population is randomly generated in the problem search space. For every particle that violates the constraints, use the gradient repair method to direct the infeasible solution toward the feasible region. In most cases, the repair method does move the solution to the feasible region. However, there may be times when the repair method fails; in such cases, the infeasible solutions are left as they are and the process continues with the next step. The constraint fitness prioritybased ranking method is then adopted to evaluate the constraint fitness value of each particle and ranks them according their constraint fitness values. The one with the greatest (constraint fitness) value computed by formulation (8) is placed at the top. If two particles have the same constraint fitness value, then their objective fitness values are compared in order to determine their relative position. The one with a better objective fitness value is positioned in front of the other. The top N + 1 particles are then fed into the NM method to improve the (N + 1) th particle. The PSO method adjusts the other 20N particles by taking into account the positions of the N + 1 best particles. This procedure for adjusting the 20N particles involves selection of the global best particle, selection of the neighbourhood best particles and, finally, velocity updates. The global best particle of the population is determined according to the sorted fitness values. Next, the 20N particles are evenly divided into 10N neighbourhoods with 2 particles per neighbourhood. The particle with the better fitness value in each neighbourhood is denoted as the neighbourhood

8 Engineering Optimization 1037 Figure 1. The hybrid NM-PSO algorithm embedded with constraint handling methods. best particle. By Equations (13) and (14), a velocity update for each of the 20N particles is then carried out. The 21N + 1 particles are sorted in preparation for repeating the entire run. The process terminates when a certain convergence criterion is satisfied. Figure 1 summarizes the algorithm of NM-PSO. For further details of the hybrid NM-PSO for unconstrained optimization problem, see Fan and Zahara (2007). 4. Experimental results In this study, the proposed NM-PSO was applied to solve 13 benchmark problems studied by Becerra and Coello (2006). This test suite is a set of difficult nonlinear functions for constrained optimization and includes four maximization problems G02, G03, G08 and G12. The characteristics of these problems include the number of decision variables (n), the type of objective function, the size of feasible region (k), the number of linear inequalities (LI), nonlinear equalities (NE) and nonlinear inequalities (NI), and the number of active constraints at the optimum (a) are summarized in Table 1. The size of the feasible region, empirically determined by simulation (Koziel and Michaelwicz 1999), indicates the degree of difficulty of randomly generating a feasible solution the greater the size, the higher the degree of difficulty. The forms of the 13 problems are described completely in Appendix 1. The algorithm is coded in Matlab 7.0 and the simulations were run on a Pentium IV 2.4 GHz with 512 MB memory capacity. Table 1. Characteristics of benchmark problems. n Type of f (x) k (%) LI NE NI a G01 13 Quadratic G02 20 Nonlinear G03 10 Polynomial G04 5 Quadratic G05 4 Cubic G06 2 Cubic G07 10 Quadratic G08 2 Non-linear G09 7 Polynomial G10 8 Linear G11 2 Quadratic G12 3 Quadratic G13 5 Exponential

9 1038 E. Zahara and C.-H. Hu During the simulation, it was found that the population size has a significant effect on the performance of NM-PSO. Table 2 shows the performance of NM-PSO with different population sizes for test problems G01, G04, G05 and G08 and it is clearly seen that for the population sizes 21N + 1 and 31N + 1, the same performance is achieved with better solutions than for 11N + 1. Therefore, a population size of 21N + 1 was employed in NM-PSO algorithm instead of 31N + 1 to decrease the number of function evaluations. The task of optimizing each of the test functions was executed 20 times by NM-PSO, and the search process was iterated 5000 times until the reference or a better solution was found. Table 3 contains a summary of the execution results of NM-PSO, and includes the worst and the best solutions obtained, the means, the medians and the standard deviations of the solutions, the average number of function evaluations, the average number of iterations and the average number of computation times. For the sake of comparison, the table also gives the reference optimal values. In every problem, the best solutions are almost equal to the reference optimal solutions. For problems G05 and G10, the optimal solutions are better than the reference. For problems G01, G03, G04, G05, G06, G08, G09, G11 and G12, optimal solutions were found consistently in all 20 runs. For problems G02 and G13, the optimal solutions were not consistently found. Finally, for problems G07 and G10, quite satisfactory global optima were found. In terms of the efficiency of NM-PSO, Table 3 shows that more than half of test problems converged in less than 300 iterations (or equivalently less than 50 s) and problems G09 and G13 converged in less than 2800 iterations (less than 240 s). However, test problems G02, G07, and G10 never completely converged; they all stopped at the pre-set maximum number of iterations (5000) with a high standard deviation, signifying that the solutions found thus far are still quite far apart. These 13 benchmark problems were solved by NM-PSO and the solutions are now compared with those obtained by applying the cultural differential evolution (CDE) algorithm (Becerra and Coello 2006), the filter simulated annealing (FSA) method (Hedar and Fukushima 2006), the genetic algorithm (GA) (Chootinan and Chen 2006) and the evolutionary algorithm (EA) (Runarsson and Yao 2005). Table 4 lists the results of solving these problems by the various methods using 20 runs in terms of the best, the mean, and the worst solutions, the standard deviation, and the average of function evaluations of each method. Solutions to problems G12 and G13 by GA are not readily available and therefore are not included in the table. Better values Table 2. The effect of population size on NM-PSO. Population size 11N N N + 1 G01 (min) Best Mean Worst Std G04 (min) Best Mean Worst Std e e 006 G05 (min) Best Mean Worst Std e e 004 G08 (max) Best Mean Worst Std e e 008

10 Table 3. Results of test problems using NM-PSO. Reference Best objective Median objective Mean objective Standard Worst objective Average no. of Average no. Average computation Type optimal value value value value deviation value func evaluations of iterations time (s) G01 Min G02 Max G03 Max G04 Min G05 Min G06 Min G07 Min G08 Max G09 Min G10 Min G11 Min G12 Max G13 Min Engineering Optimization 1039

11 1040 E. Zahara and C.-H. Hu Table 4. Comparison results of test problems with CDE, FSA, GA and ES. Best objective Mean objective Worst objective Standard Average no. of Type Algorithm value value value deviation func evaluations G01 Min CDE FSA GA EA NM-PSO G02 Max CDE FSA GA EA NM-PSO G03 Max CDE FSA GA EA NM-PSO G04 Min CDE FSA GA EA NM-PSO G05 Min CDE FSA GA EA NM-PSO G06 Min CDE FSA GA EA NM-PSO G07 Min CDE FSA GA EA NM-PSO G08 Max CDE FSA GA EA NM-PSO G09 Min CDE FSA GA EA NM-PSO G10 Min CDE FSA GA EA NM-PSO G11 Min CDE FSA GA (Continued)

12 Engineering Optimization 1041 Table 4. Continued. Best objective Mean objective Worst objective Standard Average no. of Type Algorithm value value value deviation func evaluations EA NM-PSO G12 Max CDE FSA EA NM-PSO G13 Min CDE FSA EA NM-PSO in each category are highlighted in bold type. Upon examining the results in Table 4, it is seen that NM-PSO consistently found the global optimal solutions in 10 test problems out of the 13, while the other methods found the optimum for at most 8 test problems. In terms of computational costs, NM-PSO outperformed the other four methods in 8 problems out of the 13. Considering both solution quality and computational effort, NM-PSO is superior to the CDE method in 10 test problems out of 13 and surpassed the other three methods in all 13 problems, making NM- PSO a clear winner. The superiority of NM-PSO over the other methods can be attributed to two factors. (1) NM-PSO can effectively reach a global optimum because of the mechanism of the gradient repair method, which ensures that no infeasible intermediate solution will ever enter consideration and corrupt the solution quality. The gradient repair mechanism also helps transform a constrained problem into an unconstrained one once all constraints have been satisfied. (2) The hybrid NM PSO algorithm is a proven technique (Fan and Zahara 2007) for efficiently solving unconstrained optimization problems. Additionally, NM-PSO is easy to use because it does not call for the selection of an appropriate penalty value for every test problem. It can therefore be concluded that NM-PSO is a robust and suitable tool for solving COPs. The convergence behaviour of NM-PSO falls into two general patterns. The repair method either moves an infeasible solution into the feasible region the first time the method is invoked, or it will take several iterations for the method to be effective. Solving test problems G01 and G06 will provide more insight into the convergence behaviour of NM-PSO. Figures 2 and 3 depict the relationship between the number of iterations, values of constraint fitness function and values of objective fitness function. Figure 2(a) shows the first pattern, in which the constraint fitness value reaches 1 after the repair procedure is called for the first time. Figure 2(b) shows that the objective fitness value keeps decreasing for minimization problem G01 until NM-PSO reaches the global optimum after 40 iterations. In the case of a maximization problem, the objective fitness value would continue to increase. Figure 3(a) shows the second pattern, in which the constraint fitness value does not reach 1 until the repair method is called for the eighth iteration. Before this happens, the objective fitness value would oscillate, as seen in Figure 3(b). This means that until 1 is reached by the constraint fitness function, the intermediate solution satisfies all the constraints and the objective value begins to decrease. Figures 2 and 3 convey one important message the repair method helps set NM-PSO on the right course of optimization and when infeasible solutions have been repaired and all solutions satisfy the constraints, NM-PSO can proceed quickly to the global optimum.

13 1042 E. Zahara and C.-H. Hu Figure 2. (a) Relation between iteration and constraint fitness function for G01. (b) Relation between iteration and objective fitness function for G01. It can be concluded that empowering the NM-PSO algorithm with constraint handling methods is a recommended technique for solving COPs. 5. Application to a ceramic grinding optimization problem Many engineering applications involve the use of advanced structural ceramics such as silicon carbide (SiC), silicon nitride, alumina or partially stabilized zirconium. These materials feature a high strength at elevated temperatures, resistance to chemical degradation, wear resistance and low density. The effective use of these structural ceramics in functional applications demands the grinding of ceramic components with good surface finish and low surface damage. The main concern about the use of ceramics in industry is the complexity involved in machining because of

14 Engineering Optimization 1043 Figure 3. (a) Relation between iteration and constraint fitness function for G06. (b) Relation between iteration and objective fitness function for G06. their high hardness and low fracture toughness. Therefore, optimization in ceramic processing and manufacturing technology is necessary for the commercialization of ceramic use (Lee et al. 2007). A model available in the literature (Gopal and Rao 2003) is adopted. The objective function of the optimization problem of the ceramic grinding process can be described as maximizing the material removal rate (MRR) subject to a set of constraints comprising surface roughness, number of flaws and input variables. The problem is a COP and is formulated as the objective function: Max MRR = f d c subject to R a R a(max),r a = 0.145dc f M N c N c(max),n c = 29.67dc f μm d c 30μm 8.6m/ min f 13.4m/ min 120 M 500 (15)

15 1044 E. Zahara and C.-H. Hu Table 5. Optimization results for ceramic grinding optimization problem. Constraints Value of variables Value of constraints R a(max) N c(max) d c f M R a N c Best MRR where f is the table feed rate (m/min), d c is the depth of cut (μm), R a(max) and N c(max) are the maximum allowable values of surface roughness and the number of flaws, respectively and M is grit size. Therefore, this is a three-dimensional problem whereby the values of d c, f and M are determined by NM-PSO. The NM-PSO algorithm was run for 100 trials on the same problem with a total of 9 different combinations of constraints; the search process was iterated 100 times and the results are recorded in Table 5. In all 9 cases, no violation was observed for any constraints. This means Table 6. Statistical comparison between GA, PSO and NM-PSO. Standard Number of function R a(max) N c(max) Algorithm Best MRR Mean MRR Worst MRR deviation evaluations GA PSO e NM-PSO e GA PSO e NM-PSO e GA PSO NM-PSO GA PSO e NM-PSO e GA PSO e NM-PSO e GA PSO e NM-PSO GA PSO e NM-PSO e GA PSO e NM-PSO e GA PSO e NM-PSO e

16 Engineering Optimization 1045 that NM-PSO is efficient and practical for this engineering problem. Thus, it will be more advantageous to grind SiC at a depth cut of d c = μm and feed rate f = m/min with a grit size M = mesh. For a higher range of the number of flaws, an increase of 30 to 60% in MRR can be observed by increasing the roughness value from 0.3 to 0.4 μm. This can be a good approach to achieve the specified MRR with constraints on surface roughness and surface damage. To prove the superiority and robustness of NM-PSO, the results are compared with those of GA and PSO (Gopal and Rao 2003) in Table 6. It can be observed that NM-PSO found better MRR solutions forn c(max) equal to 9 or 11. For N c(max) equal to 7, the solution is the same as that of PSO, but NM-PSO requires less function evaluations to reach the global optimum. The standard deviations associated with NM-PSO are quite small, signifying that NM-PSO converges with a high degree of accuracy. 6. Conclusions In this article, NM-PSO with embedded constraint handling methods is proposed for solving COPs. Experimental results clearly illustrate the attractiveness of this method by handling several types of constraints. It can produce competitive, if not better, solutions as compared to CDE, FSA and GA and appears to be a promising constraint handling technique. As an application, NM-PSO was employed to solve the optimization problem of silicon carbide grinding and it was proved that NM-PSO is superior in comparison with GA and PSO. In the future, NM-PSO will be applied to various real-world problems. Additionally, it would be desirable to explore the benefits of using a belief space in other types of problems. The authors are also interested in extending the approach to deal with multi-objective problems. References Becerra, R.L. and Coello, C.A., Cultured differential evolution for constrained optimization. Computer Methods in Applied Mechanics and Engineering, 195 (33 36), Chootinan, P. and Chen, A., Constraint handling in genetic algorithms using a gradient-based repair method. Computers and Operations Research, 33 (8), Coello Coello, C.A., Theoretical and numerical constraint handling techniques used with evolutionary algorithms; a survey of the state of art. Computer Methods in Applied Mechanics and Engineering, 191 (11 12), Dong, Y., Tang, J.F., Xu, B.D., and Wang, D., An application of swarm optimization to nonlinear programming. Computers and Mathematics with Applications, 49 (11 12), Eberhart, R.C. and Kennedy, J., A new optimizer using particle swarm theory. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, 4 6 October, 39 43, IEEE Service Center. Eberhart, R.C. and Shi,Y., Tracking and optimizing dynamic systems with particle swarms. Proceedings of Congress on Evolutionary Computation, Seoul, Korea, May, 94 97, IEEE Service Center. Fan, S.K. and Zahara, E., A hybrid simplex search and particle swarm optimization for unconstrained optimization. European Journal of Operational Research, 181 (2), Farmani, R. and Wright, J.A., Self-adaptive fitness formulation for constrained optimization. IEEE Transactions on Evolutionary Computation, 7 (5), Fung, R.Y.K., Tang, J.F., and Wang, D., Extension of a hybrid genetic algorithm for nonlinear programming problems with equality and inequality constraints. Computers and Operations Research, 29 (3), Gopal, A.V. and Rao, P.V., The optimization of the grinding of silicon carbide with diamond wheels using genetic algorithms. International Journal of Advanced Manufacturing Technology, 22 (7 8), Hedar, A.D. and Fukushima, M., Derivative-free filter simulated annealing method for constrained continuous global optimization. Journal of Global Optimization, 35 (4), Hu, X. and Eberhart, R.C., Tracking dynamic systems with PSO: where s the cheese? Proceedings of Workshop on Particle Swarm Optimization, Indianapolis, IN, USA Purdue School of Engineering and Technology. Koziel, S. and Michaelwicz, Z., Evolutionary algorithms, homomorphous mappings, and constrained parameter optimization. Evolutionary Computation, 7 (1),

17 1046 E. Zahara and C.-H. Hu Lee, T.S., Ting, T.O., Lin, Y.J., and Htay, T., A particle swarm approach for grinding process optimization analysis. International Journal of Advanced Manufacturing Technology, 33 (11 12), Nash, S.G. and Sofer, A., Linear and nonlinear programming. New York: McGraw-Hill. Nelder, J.A. and Mead, R., A simplex method for function minimization. Computer Journal, 7, Runarsson, T.P. andyao, X., Search biases in constrained evolutionary optimization. IEEE Transactions on Systems, Man and Cybernetics, 35 (2), Tang, J.F., Wang, D.W., Ip, A., and Fung, R.Y.K., A hybrid genetic algorithm for a type of non-linear programming problems. Computers and Mathematics with Applications, 36 (5), Appendix 1: Benchmark problems G01 Minimize f(x)= 5 x i 5 xi 2 x i i=1 i=1 i=1 g 1 (x) = 2x 1 + 2x 2 + x 10 + x g 2 (x) = 2x 1 + 2x 3 + x 10 + x g 3 (x) = 2x 2 + 2x 3 + x 11 + x g 4 (x) = 8x 1 + x 10 0 g 5 (x) = 8x 2 + x 10 0 g 6 (x) = 8x 3 + x 12 0 g 7 (x) = 2x 4 x 5 + x 10 0 g 8 (x) = 2x 6 x 7 + x 10 0 g 9 (x) = 2x 8 x 9 + x x i 1,i = 1, 2,...,9, 0 x i 100,i = 10, 11, 12, 0 x i 1,i = 13, G02 Maximize ni=1 cos 4 (x i ) 2 n f(x)= i=1 cos 2 (x i ) ni=1 ixi 2 g 1 (x) = 0.75 n i=1 x i 0 g 2 (x) = n i=1 x i 7.5n 0 n = 20 0 x i 10,i = 1,...,n G03 Maximize f(x)= ( n n) n x i i=1 n h 1 (x) = xi 2 1 = 0 i=1 n = 10 0 x i 1,i = 1,...,n

18 Engineering Optimization 1047 G04 Minimize f(x)= x x 1x x g 1 (x) = x 2 x g 2 (x) = x 2 x x 1 x x 3 x 5 0 g 3 (x) = x 2 x x 1 x x g 4 (x) = x 2 x x 1 x x g 5 (x) = x 3 x x 1 x x 3 x g 6 (x) = x 3 x x 1 x x 3 x x x x i 45,i = 3, 4, 5 G05 Minimize f(x)= 3x x x 2 + ( /3)x2 3 g 1 (x) = x 4 + x g 2 (x) = x 3 + x h 3 (x) = 1000 sin( x ) sin( x ) x 1 = 0 h 4 (x) = 1000 sin(x ) sin(x 3 x ) x 1 = 0 h 5 (x) = 1000 sin(x ) sin(x 4 x ) = 0 0 x x x i 0.55,i = 3, 4, G06 Minimize f(x)= (x 1 10) 3 + (x 2 20) 3 g 1 (x) = (x 1 5) 2 (x 2 5) g 2 (x) = (x 1 6) 2 + (x 2 5) x x G07 Minimize f(x)= x1 2 + x2 2 + x 1x 2 14x 1 16x 2 + (x 3 10) 2 + 4(x 4 5) 2 + (x 5 3) 2 + 2(x 6 1) 2 +5x (x 8 11) 2 + 2(x 9 10) 2 + (x 10 7) g 1 (x) = x i + 5x 2 3x 7 + 9x 8 0 g 2 (x) = 10x 1 8x 2 17x 7 + 2x 8 0 g 3 (x) = 8x 1 + 2x 2 + 5x 9 2x g 4 (x) = 3(x 1 2) 2 + 4(x 2 3) 2 + 2x3 2 7x g 5 (x) = 5x 1 + 8x 2 + (x 3 6) 2 2x g 6 (x) = x (x 2 2) 2 2x 1 x x 5 6x 6 0 g 7 (x) = 0.5(x 1 8) 2 + 2(x 2 4) 2 + 3x5 2 x g 8 (x) = 3x 1 + 6x (x 9 8) 2 7x x i 10,i = 1,...10

19 1048 E. Zahara and C.-H. Hu G08 Maximize f(x)= sin2 (2πx 1 ) sin(πx 2 ) x1 3(x 1 + x 2 ) g 1 (x) = x1 2 x g 2 (x) = 1 x 1 + (x 2 4) x i 10,i = 1, 2 G09 Minimize f(x)= (x 1 10) 2 + 5(x 2 12) 2 + x (x 4 11) x x2 6 + x4 7 4x 6x 7 10x 6 8x 7 g 1 (x) = x x4 2 + x 3 + 4x x 5 0 g 2 (x) = x 1 + 3x x3 2 + x 4 x 5 0 g 3 (x) = x 1 + x x2 6 8x 7 0 g 4 (x) = 4x1 2 + x2 2 3x 1x 2 + 2x x 6 11x x i 10,i = 1,...,7 G10 Minimize f(x)= x 1 + x 2 + x 3 g 1 (x) = (x 4 + x 6 ) 0 g 2 (x) = (x 6 + x 7 x 4 ) 0 g 3 (x) = (x 8 x 5 ) 0 g 4 (x) = x 1 x x x g 5 (x) = x 2 x x 5 + x 2 x x 4 0 g 6 (x) = x 3 x x 3 x x x x i 10000,i = 2, 3, 10 x i 1000,i = 4,...,8, G11 Minimize f(x)= x1 2 + (x 1) 2 h(x) = x 2 x x i 1,i = 1, 2, G12 Maximize f(x)= (100 (x 1 5) 2 (x 2 5) 2 (x 3 5) 2 )/100 g(x) = (x i p) 2 + (x 2 q) 2 + (x 3 r) x i 10,i = 5, 5, 5,

20 Engineering Optimization 1049 G13 Minimize f(x)= e x 1x 2 x 3 x 4 x 5 h 1 (x) = x1 2 + x2 2 + x2 3 + x2 4 + x2 5 h 2 (x) = x 2 x 3 5x 4 x 5 = 0 h 3 (x) = x1 3 + x = x i 2.3,i = 1, 2, 3.2 x i 3.2,i = 3, 4, 5,