Artificial Immune System Based Local Search for Solving Multi-Objective Design Problems

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1 Aerican Journal of Neural Networks and Applications 07; 3(3): doi: 0.648/.anna ISSN: (Print); ISSN: (Online) Artificial Iune Syste Based Local Search for Solving Multi-Obective Design Probles Adel M. El-Refaey Basic and Applied Science Departent, College of Engineering and Technology, Arab Acadey for Science, Technology and Maritie Transport, Cairo, Egypt Eail address: To cite this article: Adel M. El-Refaey. Artificial Iune Syste Based Local Search for Solving Multi-Obective Design Probles. Aerican Journal of Neural Networks and Applications. Vol. 3, No. 3, 07, pp doi: 0.648/.anna Received: October 4, 07; Accepted: Noveber 0, 07; Published: Deceber 4, 07 Abstract: In this paper, an artificial intelligent approach based on the clonal selection principle of Artificial Iune Syste (AIS) and local search (LS) is propose to solve Multiobective engineering design probles. This paper presents an optial design of a linear synchronous otor (LSM) considering two obective functions naely, axiu force and iniu saturation and then design of air-cored solenoid with axiu inductance and iniu volue as the obective functions. The proposed approach uses Local search, doinance principle and feasibility to identify solutions that deserve to be cloned. Keywords: Artificial Iune Syste, Local Search, Multiobective Prograing, Clonal Selection, Design Optiization. Introduction Coputing and engineering have been enriched by the introduction of the biological ideas to help developing solutions for various probles. [7] Artificial Iune Systes (AIS) are coputational paradigs that belong to the coputational intelligence faily and are inspired by the biological iune syste. [5] Clonal selection theory was proposed by Burnet (959). The theory is used to explain basic response of adaptive iune syste to antigenic stiulus. It establishes the idea that only those cells capable of recognizing an antigen will proliferate while other cells are selected against. Clonal selection operates on both B and T cells. B cells, when their antibodies bind with an antigen, are activated and differentiated into plasa or eory cells. Prior to this process, clones of B cells are produced and undergo soatic hyper utation. As a result, diversity is introduced into the B cell population. Plasa cells produce antigen-specific antibodies that are work against antigen. Meory cells reain with the host and proote a rapid secondary response. The nuber of clones for the pool of best antibodies depends on the antibody-antibody affinity. These best antibodies are selected for a unifor utation, with utation probability proportional to antibody-antigen affinity, according to the ranking schee, while the reaining population undergoes a non-unifor utation. Again, the ranking schee is used as criterion to reduce the population to its original cardinality. [, 3] Local search techniques have been very popular as heuristics for hard cobinatorial optiization probles. The basic idea is to start fro an initial solution and to search for successive iproveents by exaining neighboring solutions. The local search used in this paper is based on a dynaic version of pattern search technique. Pattern search technique is a popular paradig in Direct Search (DS) ethods [4]. The growth in deand for linear otors is principally driven by the replaceent of traditional echanical (ball screws, gear trains, cas), hydraulic, or pneuatic linear otion systes in anufacturing processes, achining, aterial handling, and positioning with direct electroechanical drives. The linear synchronous otor (LSM) operates on the sae working principle as that of a peranent agnet rotary D. C. otor [, 3]. As in a rotary otor there are two parts in a LSM, one is the set of peranent agnets and the other is the arature that has conductors carrying current. The peranent agnets produce a agnetic flux perpendicular to the direction of otion. The flow of current is in the direction perpendicular to both the direction of the otion and the direction of the agnetic flux. [] This paper intends to present an optial design of a LSM to replace a hydraulic actuator and design air-cored solenoid

2 30 Adel M. El-Refaey: Artificial Iune Syste Based Local Search for Solving Multi-Obective Design Probles. using hybrid AIS approach. This ethodology cobined AIS with Local Search, AIS find initial Pareto front then Local search iprove this initial Pareto after that applying utation process. Results show that the cobination between AIS and LS iprove the solution quality of Multiobective design optiizations and explore large area in obectives space. The reainder of the paper is organized as follows. In Section describe soe preliinaries on Multiobective optiization proble (MOP). In Section 3 review the artificial iune syste. In Section 4 explain a local search technique, in section 5 present the proposed approach. Experiental results are given and discussed in Section 6. Section 7 indicates conclusion.. Preliinaries A general Multiobective optiization proble is expressed as follows: [6] MOP: M in F ( x ) = ( f ( x ), f ( x ),..., f ( x )) s. t. x S n Where ( k ) obective functions f : R R, the decision (variable) vector x = ( x, x,..., x ) t n belongs to the (nonepty) feasible region (set) S, which is a subset of the decision n variable space R. [0] * Definition (Pareto optial solution): x is said to be a Pareto optial solution of MOP If there exists no other * feasible x (i.e., x S ) such that, f ( x ) f ( x ) for all * =,,..., k and f ( x ) < f ( x ) for at least one obective function f. [0] The ters doinance and Pareto optiality can be atheatically defined for a general proble of siultaneously iniizing a k-coponents vector function F = ( f ( x), f ( x),..., f k ( x)) of an n diensional decision variable vector x = ( x, x,..., x n ) fro soe universe S (figure ). Figure. The concept of Pareto optiality. i k t 3. Artificial Iune Systes The ain goal of the iune syste is to protect the huan body fro the attack of foreign (harful) organiss. The iune syste is capable of distinguishing between the noral coponents of organis and the foreign aterial that can cause us har (e.g. bacteria). These foreign organiss are called antigens (Ag's). The olecules called antibodies (Ab's) play the ain role on the iune syste response. The iune response is specific to a certain foreign organis (antigen). When an antigen is detected, those antibodies that best recognize an antigen will proliferate by cloning. This process is called clonal selection principle, the new cloned cells undergo high rate of utation. [6]. In Natural Iune Syste (NIS) research, four odels of the NIS can be found:. The classical view of the iune syste is that the iune syste distinguishes between self and nonself, using lyphocytes produced in the lyphoid organs. These lyphocytes learn to bind to antigen. [8]. Clonal selection theory, where an active B-Cell produces antibodies through a cloning process. The produced clones are also utated. 3. Danger theory, where the iune syste has the ability to distinguish between dangerous and nondangerous antigen. 4. Network theory, where it is assued that B-Cells for a network. When a B-Cell responds to an antigen, that B-Cell becoes activated and stiulates all other B- Cells to which it is connected in the network. [] 5. Clonal Selection Theory One exaple of a cellular evolution is the developent of the B cell (and T cell) iune repertoire. B and T cells are cells of the adaptive iune response. In contrast to the innate iune response, which is always ready to respond to whatever intruder, the adaptive iune response atures throughout life, is antigen (Ag) specific and long-living. The specificity of B cells lies in the variable region of their antibodies, each B cell produces antibodies (Ab s) with one particular specificity. [3] Ab s are olecules attached priarily to the surface of B cells whose ai is to recognize and bind to Ag s. Each B cell secretes a single type of Ab, which is relatively specific for the Ag. By binding to these Ab s and with a second signal fro accessory cells, such as the T-helper cell, the Ag stiulates the B cell to proliferate (divide) and ature into terinal (nondividing) Ab secreting cells, called plasa cells. The process of cell division (itosis) generates a clone, i.e., a cell or set of cells that are the progenies of a single cell. B cells, in addition to proliferating and differentiating into plasa cells, can differentiate into long-lived B eory cells. Meory cells circulate through the blood, lyph, and tissues and, when exposed to a second antigenic stiulus, coence to differentiate into plasa cells capable of producing high-affinity Ab s, preselected for the specific Ag that had stiulated the priary response. Figure. depicts

3 Aerican Journal of Neural Networks and Applications 07; 3(3): the clonal selection principle. [] The ain features of the clonal selection theory [, 0] that will be explored in this paper are: ) Proliferation and differentiation on stiulation of cells with Ag s. ) Generation of new rando genetic changes, expressed subsequently as diverse Ab patterns, by a for of accelerated soatic utation (a process called affinity aturation). 3) Estiation of newly differentiated lyphocytes carrying low-affinity antigenic receptors. Figure 3. Mechanis of dynaic pattern search in 5. The Proposed Approach R. 4. Local Search Figure. Clonal selection principle. The local search phase is ipleented as a dynaic version of pattern search technique. Pattern search technique is a popular paradig in Direct Search (DS) ethods. DS ethods are evolutionary algoriths used to solve constrained optiization probles. [4] This study exaines the iportance of a dynaic version of pattern search technique to iprove the solution quality of MOPs. The search procedure looks for the best solution near in the neighborhood of the current solution by repeatedly aking sall changes to a starting solution. The local search is started by loading the Pareto solutions for a given MOPs. At iteration t, an iterate xt Pareto is obtaoned, where the changes on the values for each diension ( i =,,, n ) can be ipleented as t = R r t T ( / ) ( ) where r is the rando nuber in the range [0, ]; T is the axiu nuber of iterations; R is the search radius. Let e i, ( i =,,, n ), denote the standard unit basis vectors. Successively look at the points x = x ± + t t. ei, ( i =,,, n ), until finding x + for which f ( x + ) doinates f ( x t ) for at least one obective, If the doinance not satisfied then x = + xt. Then update the Pareto solutions by non doinated ones and the doinated ones are reoved. This situation is represented in Figure 3. [] Figure 4. The proposed algorith.

4 3 Adel M. El-Refaey: Artificial Iune Syste Based Local Search for Solving Multi-Obective Design Probles. The algorith run with rando input to AIS which has taken ideas fro the clonal selection principle, [7] odeling the fact that only the highest affinity antibodies will go through local search algorith then proliferate. Antibodies, in this case, are represented by decial value which represent the value of decision variables of the proble to be solved. However, not using a population of antigens, but only Pareto doinance and feasibility to identify solutions that deserve to be cloned. Additionally, theproposed approach uses utation [8] (unifor utation is applied to the clones and non-unifor utation is applied to the not so good antibodies). Also using a secondary (or external) population that stores the nondoinated solutions found along the search process. Such secondary population is the elitist echanis ost coonly adopted in ultiobective optiization, and it allows us to ove towards the Pareto front [7]. The algorith The proposed algorith for solving ultiobective design probles using Multiobective Iune Syste Algorith (MISA) with Local search is as follow: [Step ] Rando Initialization [Step ] Sorting population according to doinance [Step 3] Choose the best antibodies to be cloned (nondoinated Antibodies) [Step 4] Apply local search for "best" antibodies to find the "best of best" Antibodies [Step 5] Cloning the best of best antibodies [Step 6] Appling a unifor utation to the clones [Step 8] Repeat this process fro step till reach the desired nuber of antibodies 6. Nuerical Results In order to validate the proposed approach, it used to solve two engineering design probles. 6.. Shape Design of a Linear Synchronous Motor A linear otor is an electric otor that has its stator and rotor "unrolled" so that instead of producing a torque (rotation) it produces a linear force along its length. To perit ore flexibility of operation and allow short headways for high-capacity operation, a design has been proposed with very short stator sections. With appropriate design, the operation control syste (signaling syste) can be integrated with the power feeding syste. [4] The task is to design a direct electrical drive actuator as an alternative to hydraulic cylinder drive. The force can be calculated using the so-called Bli law, which says that the force is the product of the flux density, the length of the conductors and the current through the conductors. To increase the force one needs to either increase the flux density, the length of the conductors or the total current. At the sae tie one has to consider the inherent as well as external liitations that appear in the for of constraints such as: [, 5] a) Heat Constraint: The preliinary calculations for the LSM show that approxiately 4000W of heat can be dissipated out of the otor with the proposed arrangeent of coils. Considering 4000W as the upper liit on the heat dissipation rate. b) Radius Constraint: Usually the geoetry and the total volue available restrict the size of the LSM. This sets a liit of the total radius of the LSM consists of the agnets, the air gap, back iron in the circuit and the conductor slots. c) Saturation Constraint: Once the iron is saturated, increasing the agnetic field strength is not useful as it will not increase the aount flux and hence the force will not increase. This sets a liit on how uch the agnetic field strength can be usefully increased. The tooth in the stator has the least cross-sectional area and will saturate first. d) Deagnetization Constraint: Very high values of the arature current will produce a very large opposing agnetic field which ay deagnetize the agnets peranently thus altering the otor perforance. This is one ore liit on the arature current. e) Miniu Force Requireent: it is required to produce a force greater than that available fro any coercially available otor, a iniu force constraint on this value can be based. Matheatical Proble Stateent [] Consider the LSM to be a three phase otor with two phases conducting at any point in tie. The conductors are assued to be copper conductors and the peranent agnets to be high density Neodiu Iron Boron agnets. The fill factor, k fill, the fraction of slot volue occupied by conductors, is assued to be equal to 0.6. Foral derivations of the atheatical expressions for the force generated in a LSM and the constraints are carried out in []. The analysis is siplified with the following assuptions. Two variables, current, i, through each conductor and the nuber of conductors, ns, in each slot, appear in three expressions, naely the force expression, the heat constraint expression and the deagnetization constraint expression. In all three expressions i and ns appear together as (i*ns). Replace these two variables by a single variable naed slot current, ins, to siplify calculations. The air gap flux density is onotonically decreasing with the air gap length. There is no advantage of increasing the air gap length and hence one would like to keep it as sall as possible. However, it is always difficult to aintain a very sall air gap especially in case of the linear otors. Assue the air gap length to be equal to which is the lowest allowed in light of anufacturing considerations. The back iron on the stator side and the over side close the flux loop as shown in Figure 5. The back iron carries high flux and is susceptible to saturation. Choose the back iron thickness such that the back iron cross-sectional area is as great as the tooth cross sectional area. Thus avoiding the tooth saturation, ensured by the saturation constraint, will surely eliinate the possibility of the back iron saturation.

5 Aerican Journal of Neural Networks and Applications 07; 3(3): Figure 5. Cross-section of approxiately a one pole pitch long section of a Tubular LSM, not to scale. The total length of the LSM consists of any agnetic poles each of length τ p, the pole pitch, which is a design paraeter. The pole pitch, τ p, and the nuber of poles, p, are related as: (Total Length = τ p *p). In a three phase otor each pole pitch length consists of three slots and three teeth as shown in Figure. 5. The force generated by the LSM, given by the Bli law, has no explicit dependency on τ p and only varies to the extent that B, l or i vary with τ p. Many sall agnets or fewer long agnets yield the sae flux density, B, and also any sall width slots or few wider slots perit the sae current. However, change in τ p affects the force indirectly, such as an increase in τ p results in an increase in the back iron thickness to avoid saturation. An increase in the back iron thickness, for a fixed radius, eans reduction in slot height resulting in less current for fixed heat or it eans thinned agnets resulting in saller B or both. Thus an increase in τ p results in a decrease in the force and hence one would like to keep τ p as sall as possible. The advantage of reducing τ p continues until τ p is so sall that the agnets (whose length ust be /3 of τ p to couple the windings which are on ) becoe so close to each other that flux leaks fro agnet to agnet. This does not happen until agnet spacing is on the order of twice the gap thickness which corresponds to a τ p well below the 5 allowed in light of anufacturing considerations. Since the nuber of poles has to be an even integer and since Total Length =.09, choose p = 40 and τ p = 7. [] The relevant design variables are the current in each slot, ins, the diensions of the slot, h s and t s, and the height of the agnet, h. In the current design there are inherent liits on the values of the variables. For exaple the height of the agnet, h, and the height of the slot, h s, are both liited by the radius of the otor. The slot thickness t s can be as large as /3rd of the pole pitch. All lengths are in eters (), the slot current ins is expressed in Aperes (A) and the flux density is expressed in Tesla (T). The units of the various constants in the equations are not shown explicitly * ins * h ( h + ( hs / )) Max f = h h Min f = t h s. t 5.9*0 * ins * h ( h + ( hs / )) g = h * t g = h + h 3* t s s 7 6.8*0 * ins( h ) g3 =.7 0 h ( h ) g 4 = 5000 f 0 s The variables liits are 0 ins < A, 0 ts 0.009, 0 hs 0.08 and 0 h Pareto Front using proposed approach is shown in figure 6a. s s

6 34 Adel M. El-Refaey: Artificial Iune Syste Based Local Search for Solving Multi-Obective Design Probles. iniu saturation axiu force x 0 5 Figure 6a. Pareto Front obtained by using proposed approach. 3.49( a N / b) L = 9 + 6( a / b) + 0( c / b) The ultiobective design proble can be cast in these ters: axiize inductance L (a, b, c) and iniize volue V (a, b, c) for given length k = 0 and cross section 6 k = 0 of the current carrying wire. Due to constraints, two variables only, e.g. a and b, can be considered; finally functions L and V becoe [4] Max Min 3.49( k / 4) 9 6( / ) 5( / ) = + a b + kk π ab f πa b k k k k 4 4π a b = + + f kk s. t g = a < 4πb 0 The variables liits are 0 a 0., 0 b 0.3, Pareto Front using proposed approach is shown in figure 8a. f x Figure 6b. Pareto front by cobining MACO and Local search []. 6.. Shape Design of an Air-Cored Solenoid A ultiobective shape optiization proble of a coreless solenoid of rectangular cross-section b c and ean radius a is tackled (figure 7) f Figure 8a. Pareto Front obtained by using proposed approach. Figure 7. Cross section of the solenoid and design variables. If current is supposed to be uniforly distributed over the cross-section, given the geoetry of the solenoid and the nuber N of turns, the inductance L [µh] can be approxiated by the following forula: Figure 8b. Pareto front by cobining MACO and Local search [].

7 Aerican Journal of Neural Networks and Applications 07; 3(3): Conclusion A hybrid ultiobective optiization algorith based on the clonal selection principle and local search have been presented. The approach is able to produce results siilar or better than those generated by other evolutionary algoriths and the Pareto optial Solution ore accurate and faster. The proposed approach uses an affinity easure to control the aount of utation to be applied to the antibodies. Affinity in this case, is defined in ters of nondoinance and feasibility. The proposed approach also uses a very siple echanis to deal with constrained test functions, and results indicate that such echanis, despite its siplicity, is effective in practice. In the two design probles the proposed approach explore large obective space that other evolutionary algoriths. References [] A. A. Mousa et al, (0), A hybrid ant colony optiization approach based local search schee for ultiobective design optiizations, Electric Power Systes Research, vol.8, pp [] A. D. Deshpande, J. R. Rinderle, (00), "Linear Electric Drive for UMM", Technical Report, Departent of Mechanical and Industrial Engineering, University of Massachusetts, Aherst. [3] A. D. Deshpande, (00), "A Study of Methods to Identify Constraint Doinance In Engineering Design Probles", M. S. Thesis, Mechanical and Industrial Engineering Departent, University of Massachusetts, Ahers, California Linear Drives, Inc. [4] A. M. Arasowan and A. O. Adewui, (04), Iproved Particle Swar Optiization with a Collective Local Uniodal Search for Continuous Optiization Probles, The Scientific World Journal, Volue 04, Article ID 7989, 3 pages. [5] C. A. Coello Coello, (00), Theoretical and nuerical constraint handling techniques used with evolutionary algoriths: A survey of the state of the art, Coputer Methods in Applied Mechanics and Engineering, vol. 9, no. /, pp [6] C. A. Coello Coello and N. Cruz Cort es, (00) "An approach to solve ultiobective optiization probles based on an artificial iune syste", in First International Conference on Artificial Iune Systes (ICARIS 00), J. Tiis and P. J. Bentley (Eds.), University of Kent at Canterbury: UK, Sept. 00, pp.. ISBN [7] C. A. Coello Coello, D. A. Van Veldhuizen, and G. B. Laont, (00), Evolutionary algoriths for solving ultiobective probles, Kluwer Acadeic Publishers, New York, ISBN [8] C. A. Coello Coello and N. Cruz Cort es, (005), "Solving ultiobective optiization probles using an artificial iune syste", Genetic Prograing and Evolvable Machines, vol. 6 p 63-90, Springer Sinece + Business Media, Inc. [9] J. F. Gieras, Z. J. Pieck, (999), "Linear Synchronous Motors: Transportation and Autoation Systes", CRC Press. [0] Kaisa M. Miettinen, (00), "Nonlinear Multiobective- Optiization", Kluwer Acadeic Publishers. [] L. Nunes de Castro and F. J. Von Zuben, (999), Artificial iune systes: Part I: Basic theory and applications, Technical Report TR-DCA 0/99, FEEC/UNICAMP, Brazil. [] L. Nunes de Castro and F. J. Von Zuben, (00), "ainet: An artificial iune network for data analysis", Data Mining: A Heuristic Approach, Idea Group Publishing, USA, pp [3] L. Nunes de Castro and F. J. Von Zuben, (00), Learning and optiization using the clonal selection principle, IEEE Transactions on Evolutionary Coputation, vol. 6, no. 3, pp [4] Marco Farina et al, (00), Cost-effective Evolutionary Strategies for Pareto Optial Front Approxiation in Multiobective Shape Design Optiization of Electroagnetic Devices, Departent of Electrical Engineering, UNIVERSITY OF PAVIA, Italy. [5] R. Filoeno Coelho, (004), "Multicriteria Optiization with Expert Rules for Mechanical Design", Dissertation, Université Libre De Bruxelles Faculté Des Sciences Appliquées. [6] Rao S. S., (009), "Engineering Optiization: Theory and Practice", (4rd ed). New York: Wiley. [7] W. F. Abd El-Wahed, E. M. Zaki, A. El-Refaey, (00), "Artificial Iune Syste based Neural Networks for Solving Multi-obective Prograing Probles", Egyptian Inforation Journal Vol. () No. pp [8] W. F. Abd El-Wahed, E. M. Zaki, A. El-Refaey, (00), "Reference Point Based Multi-Obective Optiization Using Hybrid Artificial Iune Syste", Universal Journal of Coputer Science and Engineering Technology, Vol. () No. pp: 4-30.