Pressure Vessel Design Using the Dynamic Self-adaptive Harmony Search Algorithm

Ali Kattan, Member, IEEE

Department of Information Technology, Ishik University, Erbil, Iraq

ali.kattan@ishik.edu.iq

Reem A. Alrawi, Dams & Water Resources Department

Salahaddin University, Erbil, Iraq reem_alrawi@yahoo.com

ABSTRACT Solving constrained engineering problems using metaheuristic algorithms is now a common trend whereby finding optimal solutions for such problems is regarded of high value in many industrial and engineering processes. The dynamic self-adaptive harmony search (DSHS) algorithm was proposed recently by the first author to solve some continuous and constrained problems having continuous value variables. The main merit of DSHS is its ability to dynamically set the optimization parameters based on a quality measure that is computed in every optimization cycle. Solving constrained problems is considered much harder because feasible solutions are difficult to find due to the shrunk feasible search space defined by a constrained problem. In this paper, one of the most common benchmarking constrained chemical engineering problems; namely pressure vessel design, is considered. This problem is characterized by having discrete value variables in addition to continuous value ones. A modified DSHS algorithm is proposed to handle both types of variables utilizing a simple penalty function. The proposed approach gave results that are on par with other recent metaheuristic methods addressing the same benchmarking problem. KEYWORDS evolutionary algorithms, computational intelligence, metaheuristics, optimization, harmony search 1 INTRODUCTION Evolutionary algorithms (EAs) are methods for solving unconstrained optimization problems where it becomes necessary to use an efficient constrained handling techniques when solving constrained problems [1, 2]. Several EAs that deploy metaheuristics have been proposed during

the past two decades to solve constrained problems. In comparison to conventional approaches these algorithms would usually evolve to better solutions from the available search space [3]. Regardless of the metaheuristic used in these algorithms, they all comprise of a cyclic optimization process to enhance the quality of the current solutions, referred to as population, based on some criteria [4]. The harmony search (HS) algorithm and many HS variants have been proposed to solve diverse optimization problems [5-7]. However, most of these proposed algorithms would require the setting of some important parameters prior to starting the optimization process. The selection of suitable values for these parameters is usually performed experimentally considering the problem at hand. The first author has developed an HS-based algorithm known as the dynamic self-adaptive harmony search (DSHS), for solving continuous optimization problems [8]. The same algorithm was also used to solve a constrained engineering optimization problem having continuous value variables [9]. The proposed method is characterized by having the ability to dynamically set the optimization parameters based on a quality measure, namely the best-to-worst ratio of the objective function, that is computed in every optimization cycle. In this work the DSHS is modified to handle a constrained chemical engineering optimization problem, namely pressure vessel design, which uses discrete value variables in addition to continuous ones. The pressure vessel design problem is one of the most commonly used benchmarking problems used to test the quality of solution obtained by many proposed EAs [10-18].

ISBN: 978-0-9891305-4-7 ©2014 SDIWC 11

This paper is organized as follows: Section 2 is the literature review that covers the HS algorithm and some of recent EAs that used the pressure vessel design problem. Section 3 presents the proposed method. Section 4 is the experimental results and discussion and finally the conclusions are in section 5. 2 LITERATURE REVIEW Optimization plays an important role in several engineering design problems. Basically, an optimization problem is defined as having an objective function f(x) with x ∈ ℜn , where x is an n-dimensional vector x = [x1, x2, .. xn]T. Each vector component xk, k=1, .., n is bounded by a lower and upper limits Lk ≤ xk ≤ Uk. A constrained problem has a number of equality and inequality constraints, hi(x) and gj(x) and the optimization problem is defined as in equation (1). Minimize: 𝑓 𝑥    

𝑠𝑢𝑏𝑗𝑒𝑐𝑡  𝑡𝑜    ℎ! 𝑥 = 0                  𝑖 = 1, 2,… ,𝑚𝑔! 𝑥 ≤ 0                𝑗 = 1, 2,… ,𝑝 (1)

Solving constrained optimization problems is more difficult in comparison with unconstrained optimization [19]. The difficulty stems from the fact that the feasible search space is confined with limits as imposed by the problem’s constraints. One of the most common approaches to solve such problems is to use a penalty function [1]. The amount of constraint violation is used to “punish” an infeasible solution to generate a poor fitness function value in comparison to feasible solutions. By using the penalty function the problem is changed from a constrained one to an unconstrained one [19]. The penalty approach in its simplest forms is given in equation (2), where there is a number of ways to compute the penalty term in this formula [1, 13, 20]. 𝑓 𝑥 !"#$%&' =

𝑓 𝑥                                                                            𝑖𝑓  𝑥  ∈ 𝐹𝑓 𝑥 + 𝑝𝑒𝑛𝑎𝑙𝑡𝑦 𝑥                𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (2)

Several EAs have been used recently to solve constrained engineering optimization problems such as particle swarm optimization (PSO). He and Wang [21] used a co-evolutionary PSO to manage not only the obtained solutions but also the penalty factors. The argument is that it is often difficult to set suitable penalty factors. Zahara and Kao [14] proposed constraint handling methods that include a gradient repair method and constraint fitness priority-based ranking method as a special operator to deal with satisfying constraints. Another PSO-based method was proposed by Coelho [15] in which he used new combinations of quantum-behaved PSO and Gaussian probability distribution employed in well-studied continuous optimization problems of engineering design. Many other new EAs can be found in the literature such as differential evolution (DE). DE was used to solve such problems as reflected in many recent works involving a hybrid or modified techniques [1, 17, 19] The original HS algorithm, introduced by Lee & Geem [22], is an alternative optimization technique for dynamic programming, non-linear programming and linear programming. The algorithm’s main concept is based on the improvisation process that takes place in a band of musicians. Each musician plays a note that represents one component of the harmony vector, which represents a solution vector x, having the size N. This vector represents all musician notes and the vector is associated quality measure value known as the harmony value as shown in Figure (1). This vector is analogous to N dimension solution and the harmony value represents the fitness function f(x). For a complete computational procedure of the HS algorithm refer to [22].

Figure 1. The harmony vector and harmony value

ISBN: 978-0-9891305-4-7 ©2014 SDIWC 12

Since the introduction of HS, many variants have appeared promoting some advantage over the original [7, 12, 23]. One of the main drawbacks of the original HS algorithm is that it requires statically setting a certain number of parameters prior to starting the optimization process. In addition, the algorithm exhibits sensitivity to the setting of these parameters, which would inevitably affect its ability to converge and also affects the overall algorithm’s performance. Hence the settings of these parameters would require some skill in order to obtain good results [24]. Many of the HS-based approaches have promoted a self-setting technique for some of the algorithm’s parameters and these approaches have also targeted constrained optimization for engineering problems [7, 12, 25]. A variant of HS, namely DSHS, was proposed recently by the first author and used to solve unconstrained engineering optimization problems [8]. The latter work was also adopted to address a limited constrained engineering problem having continuous value variables only [9]. The following description of DSHS algorithm is taken from the last two references above. The algorithm makes use of a quality measure that is known as the best-to-worst ratio (BtW). For every optimization cycle this ratio is computed considering the current harmony memory (HM). The HM contains a number of solutions that are stored based their harmony values (fitness), which are usually sorted. The BtW ratio considers the best and worst of these values (top and last vector of the sorted HM respectively). For minimization problems, the BtW value is a value in the range (0,1) and as given by the ratio of the current best harmony fitness value to that of the current worst harmony fitness value in HM as expressed in equation (3). A higher BtW value indicates that the quality of current HM solutions is approaching that of the current best of the HM. The pitch-adjustment-rate (PAR) value in the DSHS algorithm is adjusted dynamically based on BtW and as expressed in equation (4) and (5). PARmax is set at the value of 1.0 and PARmin is set at a small value that is greater than zero. The dynamic bandwidth value (DBW) is responsible for the pitch-adjusting process where it considers each harmony vector component separately and as

given in (6) and (7). For each dimension variable (a component of the vector solution), an ActiveBW value is computed by calculating the standard deviation of the respective HM column (with index i). The constant C in equation (6) is set to a fixed value of (C=2.0) as in [9].

𝐵𝑡𝑊 =   !(!!"#$)!(!!"#$%)

(3)

𝑚𝑠𝑙𝑜𝑝 = 𝑃𝐴𝑅!"# − 𝑃𝐴𝑅!"# (4)

𝑃𝐴𝑅 = 𝑚𝑠𝑙𝑜𝑝 ∙ 𝐵𝑡𝑊 + 𝑃𝐴𝑅!"# (5)

𝐴𝑐𝑡𝑖𝑣𝑒𝐵𝑊 𝑖 = 𝐶 ∙ 𝑆𝑡𝑑𝐷𝑒𝑣(𝑥!"! ) (6)

𝐷𝐵𝑊 𝑖 =      𝑟𝑛𝑑(−𝐴𝑐𝑡𝑖𝑣𝑒𝐵𝑊 𝑖 ,𝐴𝑐𝑡𝑖𝑣𝑒𝐵𝑊 𝑖 ) (7)

3 PROPOSED METHOD The aforementioned DSHS algorithm was adopted to solve a common constrained optimization chemical engineering problem namely pressure vessel design. Several recent EAs have used this benchmarking problem to evaluate the proposed algorithm [3, 10, 11, 13, 14, 16-19, 21]. The pressure vessel design problem is shown in Figure (2). There are two variants of this problem one consisting of four inequalities, which is considered in this work, and the other consists of six inequalities [7, 22]. The design variables depicted include the thickness of the shell Ts = x1, the thickness of the head Th = x2, the inner radius R = x3 and the length of the cylindrical section of the vessel L = x4. The thickness of the rolled steel plates is 0.0625in, hence both Ts and Th are multiples of this amount while R and L are continuous variables. The design variables’ common ranges are: 1 ≤ x1, x2 ≤ 99 and 10 ≤ x3, x4 ≤ 200.

ISBN: 978-0-9891305-4-7 ©2014 SDIWC 13

Figure 2. Pressure vessel design problem.

The two variables x1 and x2 are discrete value variables representing a multiple of the steel thickness value of 0.0625 (from 1 to 99). In the original DSHS, DBW is added to the improvised component vector value, as in equation (7). In order to comply with this constrained problem, both x1 and x2 must be a multiple of the steel thickness value in the range given. The DSHS algorithm is modified to use equation (8) and (9) to rectify the improvised value after adding DBW. The value of d in equation (8) is a rounded multiple of the steel thickness that is used in equation (9) to rectify the improvised xi component within its constrained lower and upper limits.

𝑑 = 𝑟𝑜𝑢𝑛𝑑 𝑥! ÷ 0.0625 (8) 𝑥! = min  [max[  𝐿!  , 0.0625𝑑  ], 0.0625𝑈!] (9)

There are no equality terms in this problem and thus equation (1) would include only g(x) terms. The DSHS deploys the following simple penalty function given in equation (10) whereby λ is a large number to impose an adequate penalty. 𝑝𝑒𝑛𝑎𝑙𝑡𝑦 𝑥 =  

       𝑓 𝑥 +  𝜆 max  [0,𝑔! 𝑥 ]!!!! (10)

4 EXPERIMENTAL RESULTS & DISCUSSION The proposed algorithm is applied on the pressure vessel design problem making use of the additional formulas given in equation (8) through (10). The other algorithm’s fixed settings include the value of λ = 1.0E02, a harmony memory size of HMS=50, a harmony memory consideration rate of HMCR=0.95, and a maximum improvisation count of 30,000 for which the algorithm terminates. Initialization of the HM considered random distribution. A total of 30 tests were conducted and the best-achieved results are given in Table (1) along with the best results obtained by some other recent rival EAs targeting the same problem. The obtained results are on par with other recent algorithms, differing in terms of 1.0E-01. Although the proposed method used a simple penalty function in comparison with the use of an adaptive penalty function or deploying a repair method [21], the results achieved are comparable. It would be interesting to investigate using such adaptive and/or repair approaches with the proposed DSHS. 5 CONCLUSIONS The merit of DSHS algorithm is its dynamic setting of the optimization variables in every optimization cycle depending on the quality of the solutions in the current HM. The original DSHS algorithm was modified to deal with constrained engineering problems having both and discrete as well as continuous variables. The proposed technique used a simple penalty function and was successfully applied to solve a common constrained chemical engineering optimization problem. In spite of the simple penalty function used in this work the obtained results are on par in comparison to other competitive methods. Future research should investigate using an adaptive penalty functions and repair methods that would probably enhance the obtained results. The authors are currently considering such endeavor.

ISBN: 978-0-9891305-4-7 ©2014 SDIWC 14

Table 1. Obtained results in comparison with other EA methods Method x1 x2 x3 x4 g1 g2 g3 g4 f(x) DSHS 0.8125 0.4375 42.0972084 176.652351 -2.39E-05 -0.0358926 -2.3597678 -63.347649 6059.86326 Melo 2012 0.8125 0.4375 42.09844559 176.6365595 0 -

0.035880829 0 -63.36340415 6059.714335

Kayhan 2010 0.8125 0.4375 42.0984 176.6366 N.A N.A N.A N.A 6059.7143

Coelho 2010 0.8125 0.4375 42.0984 176.6372 -8.79E-07 -3.58E-02 -0.2179 -63.3628 60.59.7208

Shen 2009 0.8125 0.4375 42.098446 176.636596 -3.40E-10 -0.035881 -0.000029 -63.363404 6059.7140

7 REFERENCES [1] W. Long, X. Liang, Y. Huang, and Y. Chen, "A hybrid

differential evolution augmented Lagrangian method for constrained numerical and engineering optimization," Computer-Aided Design, vol. 45, pp. 1562–1574, 2013.

[2] L. A. Vardhan and A. Vasan, "Evaluation of Penalty Function Methods for Constrained Optimization Using Particle Swarm Optimization," in Proceedings of the 2013 IEEE Second International Conference on Image Information Processing (ICIIP-2013), Shimla, India, 2013, pp. 487-492.

[3] V. c. V. d. Melo and G. L. C. Carosio, "Investigating

Multi-View Differential Evolution for solving constrained engineering design problems," Expert Systems with Applications, vol. 40, pp. 3370–3377, 2013.

[4] A. E. Eiben and J. E. Smith, Introduction to

Evolutionary Computing. New York: Springer, 2008. [5] Z. W. Geem, "Harmony Search Applications in

Industry," in Soft Computing Applications in Industry. vol. 226/2008, ed: Springer Berlin / Heidelberg, 2008, pp. 117-134.

[6] A. Kattan and R. Abdullah, "Training of Feed-Forward

Neural Networks for Pattern-Classification Applications Using Music Inspired Algorithm," International Journal of Computer Science and Information Security, vol. 9, pp. 44-57, 2011.

[7] M. Jaberipour and E. Khorram, "Two improved

harmony search algorithms for solving engineering optimization problems," Commun Nonlinear Sci Numer Simulat, vol. 15, pp. 3316-3331, 2010.

[8] A. Kattan and R. Abdullah, "A dynamic self-adaptive

harmony search algorithm for continuous optimization problems," Applied Mathematics and Computation, vol. 219, pp. 8542-8567, 15 April 2013 2013.

[9] A. Kattan, "A Dynamic Quality-Based Harmony Search

Algorithm for Solving Constrained Engineering Optimization Problems," presented at the 2nd International Conference on Advances in Computer Science and Engineering (CSE'2013), Los Angeles, CA, USA, 2013.

[10] C. A. C. Coello, "Use of a self-adaptive penalty approach for engineering optimization problems," Computers in Industry, vol. 41, pp. 113–127, 2000.

[11] M. Mahdavi, M. Fesanghary, and E. Damangir, "An

Improved Harmony Search Algorithm for Solving Optimization Problems," Applied Mathematics and Computation, vol. 188, pp. 1567-1579, 2007.

[12] C. M. Wang and Y.-F. Huang, "Self-adaptive harmony

search algorithm for optimization," Expert Systems with Applications, vol. 37, pp. 2826-2837, 2010.

[13] H. Shen, Y. Zhu, B. Niu, and Q. H. Wu, "An improved

group search optimizer for mechanical design optimization problems," Progress in Natural Science, vol. 19, pp. 91–97, 2009.

[14] E. Zahara and Y.-T. Kao, "Hybrid Nelder–Mead

simplex search and particle swarm optimization for constrained engineering design problems," Expert Systems with Applications, vol. 36, pp. 3880–3886, 2009.

[15] L. d. S. Coelho, "Gaussian quantum-behaved particle

swarm optimization approaches for constrained engineering design problems," Expert Systems with Applications, vol. 37, pp. 1676–1683, 2010.

[16] A. H. Kayhan, H. Ceylan, M. T. Ayvaz, and G.

Gurarslan, "PSOLVER: A new hybrid particle swarm optimization algorithm for solving continuous optimization problems," Expert Systems with Applications, vol. 37, pp. 6798–6808, 2010.

[17] D. Zou, H. Liu, L. Gao, and S. Li, "A novel modified

differential evolution algorithm for constrained optimization problems," Computers and Mathematics with Applications, vol. 61, pp. 1608–1623, 2011.

[18] D. Zou, H. Liu, L. Gao, and S. Li, "Directed searching

optimization algorithm for constrained optimization problems " Expert Systems with Applications, vol. 38, pp. 8716–8723, 2011.

[19] V. c. V. d. Melo and G. L. C. Carosio, "Evaluating

differential evolution with penalty function to solve constrained engineering problems," Expert Systems with Applications, vol. 39, pp. 7860–7863, 2012.

ISBN: 978-0-9891305-4-7 ©2014 SDIWC 15

[20] T. Takahama and S. Sakai, "Constrained Optimization by Applying the alpha Constrained Method to the Nonlinear Simplex Method With Mutations," IEEE Transactions on Evolutionary Computation, vol. 9, pp. 437-451, 2005.

[21] Q. He and L. Wang, "An effective co-evolutionary

particle swarm optimization for constrained engineering design problems," Engineering Applications of Artificial Intelligence vol. 20, pp. 89–99, 2007.

[22] K. S. Lee and Z. W. Geem, "A New Meta-heuristic

Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice," Computer Methods in Applied Mechanics and Engineering, vol. 194, pp. 3902-3933, 2005.

[23] M. Fesanghary, M. Mahdavi, M. Minary-Jolandan, and

Y. Alizadeh, "Hybridizing harmony search algorithm with sequential quadratic programming for engineering optimization problems," Computer methods in applied mechanics and engineering, vol. 197, pp. 3080-3091, 2008.

[24] Z. W. Geem and K.-B. Sim, "Parameter-setting-free

harmony search algorithm," Applied Mathematics and Computation, vol. 217, pp. 3881-3889, 2010.

[25] Q. K. Pan, P. N. Suganthan, M. F. Tasgetiren, and J. J.

Liang, "A self-adaptive global best harmony search algorithm for continuous optimization problems," Applied Mathematics and Computation, vol. 216, pp. 830-848, 2010.

ISBN: 978-0-9891305-4-7 ©2014 SDIWC 16