Whale swarm algorithm for function optimization

Transcription

1 Whale swarm algorthm for functon optmzaton Bng Zeng School of Mechancal Scence and Engneerng Huazhong Unversty of Scence and Technology Wuhan, Chna Lang Gao* School of Mechancal Scence and Engneerng Huazhong Unversty of Scence and Technology Wuhan, Chna Xnyu L School of Mechancal Scence and Engneerng Huazhong Unversty of Scence and Technology Wuhan, Chna lxnyu@hust.edu.cn Abstract Increasng nature-nspred metaheurstc algorthms are appled to solvng the real-world optmzaton problems, as they have some advantages over the classcal methods of numercal optmzaton. Ths paper has proposed a new nature-nspred metaheurstc called Whale Swarm Algorthm for functon optmzaton, whch s nspred by the whales behavor of communcatng wth each other va ultrasound for huntng. The proposed Whale Swarm Algorthm has been compared wth several popular metaheurstc algorthms on comprehensve performance metrcs. Accordng to the expermental results, Whale Swarm Algorthm has a qute compettve performance when compared wth other algorthms. Keywords whale Swarm Algorthm; ultrasound; naturenspred; metaheurstc; real-world optmzaton problems; functon optmzaton. I. INTROUCTION Nature-nspred algorthms are becomng powerful n solvng numercal optmzaton problems, especally the NP-hard problems such as the travellng salesman problem [], vehcle routng [], classfcaton problems [3], routng problem of wreless sensor networks(wsn) [4] and multprocessor schedulng problem [5], etc. These realworld optmzaton problems often probably come wth multple global or local optma of a gven mathematcal model (.e., objectve functon). And f a pont-by-pont classcal method of numercal optmzaton s used for ths task, the classcal method has to try many tmes for locatng dfferent optmal soluton n each tme [6], whch wll take a lot of tme and work. Therefore, usng nature-nspred metaheurstc algorthms to solve these problems has become a hot research topc, as they are easy to mplement and can converge to the global optma wth hgh probablty. In ths paper, we have proposed a new nature-nspred metaheurstc called Whale Swarm Algorthm (WSA) for functon optmzaton, based on the whales behavor of communcatng wth each other va ultrasound for huntng. Here, a bref overvew of the nature-nspred metaheurstc algorthms s presented. Genetc Algorthm (GA) was ntally proposed by Holland to solve the numercal optmzaton problem [7], whch smulates arwn's genetc choce and natural elmnaton bology evoluton process and has opened the prelude of nature-nspred metaheurstc algorthms. It manly utlzes selecton, crossover and mutaton operatons on the ndvduals (chromosomes) to fnd the global optmum as far as possble. In GAs, the crossover operator that s utlzed to create new ndvduals by combnng parts of two ndvduals sgnfcantly affects the performance of a genetc system [8]. Untl now, lots of researchers have proposed dverse crossover operators for dfferent optmzaton problems. For nstance, Syswerda has proposed order based crossover operator (OBX) for permutaton encodng when dealng wth schedule optmzaton problem [9]. A detaled revew of crossover operators for permutaton encodng can be seen from reference []. Mutaton s another mportant operator n GAs, whch provdes a random dversty n the populaton [], so as to prevent premature convergence of algorthm. Mchalewcz has proposed random (unform) mutaton and non-unform mutaton [] for numercal optmzaton problems. And polynomal mutaton operator proposed by eb s one of the most wdely used mutaton operator [3]. A comprehensve ntroducton to mutaton operator can be seen from [4]. In a word, t s very mportant to choose or desgn approprate select, crossover and mutaton operators of GAs, when dealng wth dfferent optmzaton problems. Storn and Prce proposed fferental Evoluton (E) algorthm for mnmzng possbly nonlnear and nondfferentable contnuous space functons [5]. It also contans three key operatons, namely mutaton, crossover and selecton, whch are dfferent from those of GAs. Frst of all, a donor vector, correspondng to each member vector of the populaton called target vector, s generated n the mutaton phase of E. Then, the crossover operaton takes place between the target vector and the donor vector, wheren a tral vector s created by selectng components from the donor vector or the target vector wth the crossover probablty. The selecton process determnes whether the target or the tral vector survves n the next generaton. If the tral vector s better, t replaces the target vector; otherwse remanng the target vector n the populaton. Snce put forward, E algorthm has ganed ncreasng popularty from researchers and engneers n solvng lots of real-world optmzaton problems [6, 7] and varous schemes have been proposed for t [8]. The general conventon used to name the dfferent E schemes s E/x/y/z, where E represents fferental Evoluton, x stands for a strng ndcatng the base vector need to be perturbed, for example, t can be set as best and rand, y denotes the number of dfference vectors used to perturb x, and z represents the type of crossover operaton whch can be bnomal (bn) or exponental (exp) [8]. Some popular exstng E schemes are E/best//bn, E/best//exp, E/rand//bn, E/best//exp, E/rand//exp, etc.

2 Partcle Swarm Optmzaton (PSO) s a swarm ntellgence based algorthm proposed by Kennedy and Eberhart, whch s nspred by socal behavor of brd flockng [9]. PSO algorthm has been appled to solve lots of complex and dffcult real-world optmzaton problems [, ], snce t was put forward. In the tradtonal PSO algorthm, each partcle moves to a new poston based on the update of ts velocty and poston, where the velocty s concerned wth ts cogntve best poston and socal best poston. Untl now, there are lots of PSO varants are proposed for dfferent optmzaton problems. For nstance, Sh and Eberhart has ntroduced a lnear decreasng nerta weght nto PSO (PSO-LIW) [], whch can balance the global search and local search, for functon optmzaton. Zhan et al. have proposed Adaptve PSO (APSO) [3] for functon optmzaton, whch enables the automatc control of parameters to mprove the search effcency and convergence speed, and employs an eltst learnng strategy to jump out of the lkely local optma. Qu et al. have proposed stance-based Locally Informed PSO (LIPS) that elmnates the need to specfy any nchng parameter and enhance the fne search ablty of PSO for multmodal functon optmzaton [6], etc. In addton to the above, there are large amounts of other nature nspred algorthms such as Ant Colony Optmzaton (ACO) [4], Bees Swarm Optmzaton (BSO) [5] and Bg Bang-Bg Crunch (BB-BC) [6], etc. A comprehensve revew of the nature nspred algorthms s beyond the scope of ths paper. A detaled and complete reference on the motf can be seen from [7, 8]. The rest of ths paper s organzed as follows. Secton descrbes the proposed WSA n suffcent detal. The experment setup s presented n Secton 3. Secton 4 presents the expermental results performed to evaluate the proposed algorthm. The last secton s the conclusons and topcs for further works. II. WHALE SWARM ALGORITHM Frst of all, ths secton ntroduces the behavor of whales probably, especally the behavor of whales huntng. Then, the detals of Whale Swarm Algorthm are presented. A. Behavor of whales Whales wth great ntellectual and physcal capactes are completely aquatc mammals, and there are about eghty whale speces n the vast ocean. They are socal anmal and lve n groups. Such as pregnant females wll gather together wth other female whales and calves so as to enhance defense capabltes. And sperm whales are often spotted n groups of some 5 to ndvduals, as shown n Fg.. The whale sounds are beautful songs n the oceans and ther sound range s very wde. Untl now, scentsts have found 34 speces of whale sounds, such as whstlng, squeakng, groanng, longng, roarng, warblng, clckng, buzzng, churrng, conversng, trumpetng, cloppng and so on. These sounds made by whales can often be lnked to mportant functons such as ther mgraton, feedng and matng patterns. What s more, a large part of sounds made by whales are ultrasound whch are beyond the scope of human hearng. And whales determne foods azmuth and keep n touch wth each other from a great dstance by the ultrasound. Fg.. The swarm of sperm whales. When a whale has found food source, t wll make sounds to notfy other whales nearby of the qualty and quantty of food. So each whale wll receve lots of notfcatons from the neghbors, and then move to the proper place to fnd food based on these notfcatons. The behavor of whales communcatng wth each other by sound for huntng nspre us to develop a new metaheurstc algorthm for functon optmzaton problems. In the rest of ths secton, we wll dscuss the mplementaton of Whale Swarm Algorthm n detal. B. Whale swarm algorthm To develop whale swarm nspred algorthm for solvng functon optmzaton problem, we have dealzed some huntng rules of whale. For smplcty n descrbng our new Whale Swarm Algorthm, the followng four dealzed rules are employed: ) all the whales communcate wth each other by ultrasound n the search area; ) each whale has a certan degree of computng ablty to calculate the dstance to other whales; 3) the qualty and quantty of food found by each whale are assocated to ts ftness; 4) the movement of a whale s guded by the nearest one among the whales that are better (judged by ftness) than t, such nearest whale s called the better and nearest whale n ths paper. ) Iteratve equaton As we know, both rado wave and lght wave are electromagnetc waves, whch can propagate wthout any medum. If propagatng n water, they wll attenuate quckly due to the large electrcal conductvty of water. Whereas, sound wave s one knd of mechancal wave that needs a medum through whch to travel, whether t s water, ar, wood or metal. And ultrasound belongs to sound wave, whose transmsson speed and dstance largely depends on the medum. For nstance, ultrasound travels about 45 meters per second n water, whch s faster than that (about 34 meters per second) n ar. What s more, some ultrasound wth pre-specfed ntensty can travel about meters underwater, but can only transmt meters n ar. That s because the ntensty of mechancal wave s contnuously attenuated by the molecules of the medum, and the ntensty of ultrasound travelng n ar s attenuated far more quckly than that n water. The ntensty ρ of the ultrasound at any dstance d from the source can be formulated as follows [9]. d e where, s the ntensty of ultrasound at the orgn of source, e denotes the natural constant. η s the attenuaton

3 coeffcent, whch depends on the physco-chemcal propertes of the medum and on the characterstcs of the ultrasound tself (such as the ultrasonc frequency) [9]. As we can see from Eq., ρ decreases exponentally wth the ncrement of d when η s constant, whch means that the dstorton of message conveyed by the ultrasound transmtted by a whale wll occur wth a great probablty, when the travel dstance of the ultrasound gets qute far. So a whale wll not sure whether ts understandng of the message send out by another whale s correct, when that whale s qute far away from t. Thus, a whale would move negatvely and randomly towards ts better and nearest whale whch s qute far away from t. Based on the above, t can be seen that a whale would move postvely and randomly towards ts better and nearest whale whch s close to t, and move negatvely and randomly towards that whale whch s qute far away from t, when huntng food. Thus, some whale swarms wll form after a perod of tme. Each whale moves randomly towards ts better and nearest whale, because random movement s an mportant feature of whales behavor, lke the behavor of many other anmals such as ant, brds, etc., whch s employed to fnd better food. These rules have nspred us to fnd a new poston teratve equaton, wshng the proposed algorthm to avod fallng nto the local optma quckly and enhance the populaton dversty and the global exploraton ablty, as well as contrbute to locatng multple global optma. Then, the random movement of a whale X guded by ts better and nearest whale Y can be formulated as follows. dx, Y x x rand, e y x t+ t t t where, and are the -th elements of X s poston at t and t+ teratons respectvely, smlarly, denotes the - th element of Y s poston at t teraton. represents the t x t+ x d, Eucldean dstance between X and Y. And rand, e X Y d, means a random number between and e X Y. Based on a large number of experments, can be set to for almost all the cases. As mentoned prevous, the attenuaton coeffcent η s dependent on the physco-chemcal propertes of the medum and on the characterstcs of the ultrasound tself. Here, for functon optmzaton problem, those factors that affect η can be assocated to the characterstcs of the objectve functon, ncludng the functon dmenson, range of varables and dstrbuton of peaks. Therefore, t s mportant to set approprate η value for dfferent objectve functon. For engneer s convenence n applcaton of WSA, the ntal approxmate value of η can be set as follows, based on a large number of expermental results. Frst of all, dmax we should make e.5,.e., dmax e.5, snce s always set to, wheren d max denotes the maxmum dstance between any two whales n the search space that can be formulated as n d x x max U L, n s the dmenson of the objectve functon, L U x and x represent the lower lmt and upper lmt of the -th varable respectvely. Ths equaton means that f the dstance between whale X and ts better and nearest whale Y s d max, the part d, e XY of Eq. that affects the d X, Y t y movng range of whale X should be set to.5. Next, we can get that ln.5 dmax. Then, t s easy to adjust η to the optmal or near-optmal value based on ths ntal approxmate value. Eq. shows that a whale wll move towards ts better and nearest whale postvely and randomly, f the dstance between them s small. Otherwse, t wll move towards ts better and nearest whale negatvely and randomly, whch can be llustrated wth Fg. when the dmenson of the objectve functon s equal to. In Fg., the red stars denote the global optma, the crcles represent the whales and the rectangular regons sgned wth magnary lnes are the reachable regons of the whales n current teraton. X Y Y e e d.6 d, X, Y X Y X.4 Fg.. Sketch map of a whale s movement guded by ts better and nearest whale. ) General framework of WSA Based on the above rules, the general framework of the proposed WSA can be summarzed as shown n Fg. 3, where Ω n lne 6 denotes the number of members n Ω, namely the swarm sze, and Ω n lne 7 s the -th whale n Ω. It can be seen from Fg. 3 that those steps before teratve computaton are some ntalzaton steps, ncludng ntalzng confguraton parameters, ntalzng ndvduals postons and evaluatng each ndvdual, whch are smlar wth most other metaheurstc algorthms. Here, all the whales are randomly assgned to the search area. Next come the core step of WSA: whales move (lnes 5-3). Each whale needs to move for the better food va group cooperaton. Frst of all, a whale should fnd ts better and nearest whale (lnes 7), as shown n Fg. 4, where f(ω) n lne 6 s the ftness value of the whale Ω and dst(ω, Ωu) n lne 7 denotes the dstance between Ω and Ωu. If ts better and nearest whale exsts, then t wll move under the gudance of the better and nearest whale (lnes 9 n Fg. 3). As descrbed above, the framework of WSA s farly smple, whch s convenent for applyng WSA n solvng the realworld optmzaton problems. The general framework of Whale Swarm Algorthm Input: An objectve functon, the whale swarm Ω. Output: The global optma. : begn : Intalze parameters; 3: Intalze whales postons; 4: Evaluate all the whales (calculate ther ftness); 5: whle termnaton crteron s not satsfed do 6: for = to Ω do 7: Fnd the better and nearest whale Y of Ω ; 8: f Y exsts then 9: Ω moves under the gudance of Y accordng to Eq. ; : Evaluate Ω ; : end f : end for

4 3: end whle 4: return the global optma; 5: end Fg. 3. The general framework of WSA. The pseudo code of fndng a whale s better and nearest whale Input: The whale swarm Ω, a whale Ω u. Output: The better and nearest whale of Ω u. : begn : efne an nteger varable v ntalzed wth ; 3: efne a float varable temp ntalzed wth nfnty; 4: for = to Ω do f u then f f(ω )<f(ω u) then f dst(ω, Ω u)<temp then v=; temp=dst(ω, Ω u); end f end f end f 5: 6: 7: 8: 9: : : : 3: end for 4: return Ω v; 5: end Fg. 4. The pseudo code of fndng a whale s better and nearest whale. III. EXPERIMENT SETUP A. Expermental confguraton The proposed WSA and other algorthms compared are all mplemented wth C++ programmng language by Mcrosoft vsual studo 5 and executed on the PC wth.3 GHz Intel core 7 36QM processor, 8 GB RAM and Mcrosoft Wndows operatng system. In addton to the GA wth elte selecton, non-unform arthmetc crossover and basc bt mutaton strateges, E/best//bn [3] and PSO wth nerta weght [], the followng 4 popular multmodal optmzaton algorthms are also compared wth WSA: The locally nformed PSO (LIPS) [6], Specatonbased E (SE) [3], The orgnal crowdng E (CE) [3] and Specaton-based PSO (SPSO) [33]. In ths paper, we utlze the evaluaton number of objectve functon as the stoppng crteron of these algorthms to test ther performance. B. Test functons To verfy the performance of the proposed WSA, the comparatve experments were conducted on twelve benchmark test functons, whch are taken from the studes of eb [34], Mchalewcz [], L [35] and Thomsen [3]. Test functons F-F8 are multmodal (F-F6 and F7-F are low and hgh dmensonal multmodal functons respectvely) that have multple global or local optma. F and F are hgh dmensonal unmodal functons wth dmenson. Basc nformaton of these test functons are summarzed n Table. For functons F-F6, the objectve s to locate all the global optma, whle for the rest the target s to escape the local optma (f they have) to hunt for the global optmum. And all test functons are mnmzaton problems. It can be seen from Table, F7, F9, F and F all get the global optma at (,,, ), and F gets the global optmum at (,,, ), whch are located near the mddle of the feasble regon. As we know, some algorthms are effcent n optmzng the functons whose optma are near the mddle of the feasble regon, especally near zero, but perform badly when the optma are not near the mddle of the feasble regon. For the sake of farness, we have shfted F7-F, and the shft data of them are randomly generated wthn specfed range. Table. Test functons. No. Functon name / mensons Expresson Ranges No. of global optma Uneven Increasng Mnma / x.8 6 f X explog sn 5 x.5 [, ] /4 Uneven Mnma / f sn 5 x.5 3 Hmmelblau s functon / f x x x x Mnmum value X [, ] 5-4 Sx-hump camel back / X X 7 [-6, 6] 4 - x f x x xx x x nverted Shubert functon / j 6 Brann RCOS / f X x x x x [-.9,.9] [-.,.] f X j cos j x j [-, ] cos Rastrgn / f x x 8 Schwefel / f xsn x [-5, ] [, 5] X cos [-, ] X [-5, 5] 9 Grewank / f X cos [-, ] x x 4 Rosenbrock / f x x x Sphere / X [-5, 5] f X x [-, ] Zakharov / f X x x x [-5, ] 4

5 C. Parameters settng Although the global optma of these test functons can be obtaned by the method of dervaton, they should stll be treated as black-box problems,.e., the known global optma of these test functons cannot be used by the algorthms durng the teratons, so as to compare the performance of these algorthms. The ftness value and the number of global optma of each functon have been lsted n Table. Here, we use ftness error Ɛ f,.e., level of accuracy, to judge whether a soluton s a real global optmum,.e., f the dfference between the ftness of a soluton and the known global optmum s lower than Ɛ f, ths soluton can be consdered as a global optmum. In our experments, the ftness error Ɛ f, populaton sze and maxmal number of functon evaluatons for WSA and the 7 algorthms compared are lsted n Table. It s worth to note that a functon whch has more optma or hgher dmenson requres a larger populaton sze and more number of functon evaluatons. Functon Table. Test functons settng. Ɛ f populaton No. of functon no. sze evaluatons F. F. F3.5 F4. F5.5 3 F6. F7. 5 F8. 5 F9. 5 F. 5 F. 5 F. 5 The user-specfed control parameters of WSA,.e., attenuaton coeffcent η, for these test functons are set as shown n Table 3. Table 3. Parameter settng of WSA for test functons. Parameter F F F3 F4 F5 F6 F7 F8 F9 F F F η E-3.E-3 5E-3 6.5E- 5E-3 6.5E- The parameters value of the 7 algorthms compared are set as the same as those n ther reference source respectvely. Table 4 has shown the settng of the man parameters of these algorthms. The parameter speces radus r s of SE and SPSO for these test functons are lsted n Table 5. Algorthms Table 4. Settng of the parameters of algorthms. Parameters GA P c =.95, P m =.5 E P c =.7, F =.5 PSO ω =.79844, c =, c = LIPS ω =.79844, nsze = ~5 SE P c =.9, F =.5, m = CE P c =.9, F =.5, CF = populaton sze SPSO χ =.79844, φ =.5, φ =.5. Pc: crossover probablty; Pm: mutaton probablty;. F: scalng factor; 3. ω: nerta weght; c, c: acceleraton factor; 4. nsze: neghborhood sze; 5. m: speces sze; 6. CF: crowdng factor; 7. χ: constrcton factor; φ, φ: coeffcent. Table 5. Speces radus settng for test functons. F F F3 F4 F5 F6 F7 F8 F9 F F F SE SPSO Performance metrcs To compare the performance of WSA wth the 7 algorthms, we have conducted 5 ndependent runs for each algorthm on each test functon. And the followng fve metrcs are used to measure the performance of all the algorthms. ) Success Rate (SR) [33]: the percentage of runs n whch all the global optma are successfully located usng the gven level of accuracy. ) Average Number of Optma Found (ANOF) [36]: the average number of global optma found over 5 runs. 3) Maxmum Peak Rato Statstc (MPR) [36]: ths paper also adopts MPR to compare the qualty of optma found by dfferent algorthms. MRP s expressed as follows. MPR= q = q = * F F + * f F + where q s the number of optma found by the algorthm, q q f are the ftness value of these optma, F are the value of real optma correspondng to those optma found by the algorthm, whle F * s the value of the global optmum. It s obvous that the larger the MPR value s, the better the algorthm performs. The maxmal MPR value s. 4) Convergence speed: the speed of an algorthm convergng to the global optmum over functon evaluatons. IV. EXPERIMENTAL RESULTS AN ANALYSIS Ths secton presents and analyzes the results of comparatve experments. All the algorthms were run under the experment setup shown n the prevous secton. A. Success rate The success rates of all the algorthms on each test functon are presented n Table 6, n whch the numbers wthn parentheses denote the ranks of each algorthm. If the success rates of any two algorthms on a test functon are equal, they have the same ranks over ths test functon. The

6 last row of ths table shows the total ranks of algorthms, whch are the summaton of the ndvdual ranks on each test functon. As we can see from Table 6, for multmodal functons F-F, the success rate of WSA on F3 s only a lttle bt lower than that of LIPS, but s far greater than those of other algorthms. Only two multmodal optmzaton algorthms (.e., LIPS and SE) can acheve nonzero success rates on F5, and no algorthm can acheve nonzero success rates on the four hgh dmensonal multmodal functons F7-F. What s more, WSA has acheved % success rates on test functons F, F, F4 and F6, whch are much hgher than those ganed by most of other algorthms. Therefore, t can be seen that WSA has a very compettve performance on dealng wth multmodal functons wth respect to other algorthms. And for hgh-dmensonal unmodal functons F-F, all the algorthms cannot acheve nonzero success rates on F. However, WSA has acheved % success rate on F, whle the success rates of other algorthms on F are. Therefore, t can be concluded that WSA also has better performance than other algorthms on success rate when solvng unmodal functons. It also can be seen that the better performance of WSA on success rate can be supported by the total rank of WSA that s 5 whch s much smaller than those ganed by other algorthms. The better performance of WSA s due to ts novel teraton rules based on the behavor of whales huntng, ncludng that the random movement of a whale s guded by ts better and nearest whale, and ts range of movement depends on the ntensty of the ultrasound receved as shown as Eq., whch have a great contrbuton to the mantenance of populaton dversty and the enhancement of global exploraton ablty, so as to locate the global optmum(optma). Table 6. SR and ranks (n parentheses) of algorthms for test functons. Functon no. F F F3 F4 F5 F6 F7 F8 F9 F F F Total rank WSA GA E PSO CE SE SPSO LIPS As some algorthms cannot obtan nonzero success rates on some multmodal functons, the metrc ANOF has been used to test the performance of those algorthms on locatng multple global optma. Table 7 has presented the ANOF of all the algorthms over functons F-F6 whch have multple global optma. As can be seen from ths table, for these multmodal functons, the ANOF of WSA on test functons F, F4 and F6 are much hgher than those obtaned by most of other algorthms, whch echoes the % success rates of WSA on these functons as shown n Table 6. And the ANOF of WSA on F3 s only a lttle bt lower than that of LIPS, but s much hgher than those of other algorthms, whch s smlar to the case of success rates of algorthms on ths functon. As can be seen from Table 6, only LIPS and SE can acheve nonzero success rates on F5. Here, the ANOF of WSA on F5 s 6.76, whch s much hgher than those of other algorthms but the multmodal optmzaton algorthms LIPS and SE. Therefore, the results of Table 7 have further demonstrated the outstandng performance of WSA on fndng multple global optma wth respect to other algorthms when solvng multmodal functons. Table 7. ANOF and ranks (n parentheses) of algorthms for test functons F-F6. Functon no. WSA GA E PSO CE SE SPSO LIPS F F F F F Total rank B. Qualty of optma found MPR s utlzed to measure the qualty of optma found by algorthms. The mean and standard devaton of MPR of all the algorthms on each test functon over 5 runs are lsted n Table 8. Here, the ranks of algorthms are based on the mean of MPR over the test functons. As we can see from ths table, for low-dmensonal multmodal functons F-F6, WSA has acheved the best MPR on F and F. And E algorthm ranks the best on F3-F6. But E algorthm has not got nonzero success rates on F3-F6 as shown n Table 6, and has ganed worse ANOF than most of other algorthms as shown n Table 7, whch mean that E algorthm has a poor performance on locatng multple global optma though t can acheve a few of the multple global optma wth hgh accuracy, when solvng lowdmensonal multmodal functons. Whereas, WSA has acheved very good MPR over F3-F6, on the premse of keepng excellent SR and ANOF as shown n Table 6 and Table 7. And for hgh-dmensonal multmodal functons F7 and F8, WSA only performs a lttle bt worse than LIPS and SE, but outperforms other algorthms. What s more, for hgh-dmensonal multmodal functons F9-F and hghdmensonal unmodal functons F-F, WSA has acheved the best MPR values when compared wth all the other algorthms. Partcularly, WSA has ganed the maxmal MPR value (.e., ) on F, and the mean of the optmal value found by WSA on F over 5 runs s.6e-9 n the experments, whch s far better than those obtaned by all the other algorthms. Furthermore, the standard devatons of MPR of WSA on these test functons are qute small. Therefore, t can be concluded that WSA also performs

7 Average ftness value better than most of other algorthms n terms of the qualty of optma. The outstandng performance of WSA on the qualty of optma s also due to ts novel teraton rules, whch contrbute sgnfcantly to enhancng the local explotaton ablty. Table 8. MPR and ranks (n parentheses) of algorthms for test functons. Functon no. Measure WSA GA E PSO CE SE SPSO LIPS F F F3 F4 F5 F6 F7 F8 F9 F F F E E E E-4.77e-4 5.7e-5 4.5e-5.3e e-3.e- 8.3e e-9 3.7e e E-3 9.8E E E E-.987.E- 6.4e-6.37e-7.74e-5.4e-6.84e- 7.36e-4.8e-8.e-9 5.6e-6.84e-7.8e-3 5.e E e-5 9.3e-6 3.e e-7.e-.74e-.7e-7 7.7e e-5.e-5 8.e e E E E E-4 7.7e e-7 4.3e-5 4.7e-6.47e- 3.7e e-8 3.6e-8 7.e e e e E E E E E E-.65e e-7.6e-5 4.5e-7 7.8e-.87e e-7.37e-7.79e e e-4 3.3e E E E E E-4 5.9e-4.85e-5 4.3e-5.78e e-.4e-3.e e-6.37e-3 7.3e-5.63e-3.44e E E E E E E-.64e-4.9e e-5.63e-6.99e-.3e-.54e e e e e e-4 Total rank E E E E E E-3 6.4e-4.9e-4 5.8e-5.49e e-.39e-.6e e e e-7.78e-3.96e-4 C. Effcency Based on the prevous, t can be seen that WSA has a qute compettve performance when compared wth other algorthms, n terms of the locaton of multple global optma and the qualty of optma. Ths subsecton dscusses the effcency of all the algorthms, manly focus on the convergence speed. ) Convergence speed To further demonstrate the superorty of WSA, t s compared wth other algorthms on F3 n terms of convergence speed n ths subsecton. The convergence curves of all the algorthms on F3 are depcted n Fg. 5, n whch the abscssa values denote functon evaluatons and the ordnate values represent the average ftness values of populaton over 5 runs. As can be seen from Fg. 5, WSA converges slower than E, LIPS and SE n the early teratons. However, n the md and later teratons, WSA converges faster than LIPS and SE, and t can acheve a better value than LIPS and SE do. Although E algorthm can converge to the global optmum (-), t can only locate one of the four global optma n a sngle run, as shown n Table 7. Therefore, t can be concluded that WSA has better performance n terms of convergence speed than other algorthms on the premse of keepng good SR and ANOF. The excellent performance of WSA on convergence speed s also due to ts novel teraton rules based on the behavor of whales huntng as shown as Eq FEs GA E PSO CE SE SPSO LIPS WSA Fg. 5. The convergence graph of dfferent algorthms on F3.

8 V. CONCLUSIONS A new swarm ntellgence based metaheurstc called Whale Swarm Algorthm, nspred by the whales behavor of communcatng wth each other va ultrasound for huntng, s proposed for functon optmzaton n ths paper. The nnovatons of the teratve equaton of WSA consst of two parts: the random movement of a whale s guded by ts better and nearest whale; and ts range of movement depends on the ntensty of the ultrasound receved, whch contrbute sgnfcantly to the mantenance of populaton dversty, the avodance of fallng nto the local optma quckly and the enhancement of global exploraton ablty, so as to locate the global optmum(optma). And the novel teraton rules also have a great contrbuton to the enhancement of local explotaton ablty, especally when some whales have gathered around a same peak, so as to mprove the qualty of optma. WSA has been compared wth several popular metaheurstc algorthms on four performance metrcs (.e., SR, ANOF, MPR and Convergence speed). 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