A Hybrid Differential Evolution Algorithm Game Theory for the Berth Allocation Problem

Transcription

1 A Hybrd Dfferental Evoluton Algorthm ame Theory for the Berth Allocaton Problem Nasser R. Sabar, Sang Yew Chong, and raham Kendall The Unversty of Nottngham Malaysa Campus, Jalan Broga, Semenyh, Selangor, Malaysa Abstract. The berth allocaton problem (BAP) s an mportant and challengng problem n the martme transportaton ndustry. BAP can be defned as the problem of assgnng a berth poston and servce tme to a gven set of vessels whle ensurng that all BAP constrants are respected. The goal s to mnmze the total watng tme of all vessels. In ths paper, we propose a dfferental evoluton (DE) algorthm for the BAP. DE s a nature-nspred metaheurstc that has been shown to be an effectve method to addresses contnuous optmzaton problems. It nvolves a populaton of solutons that undergo the process of selecton and varaton. In DE, the mutaton operator s consdered the man varaton operator responsble for generatng new solutons. Several mutaton operators have been proposed and they have shown that dfferent operators are more sutable for dfferent problem nstances and even dfferent stages n the search process. In ths paper, we propose an enhanced DE that utlzes several mutaton operators and employs game theory to control the selecton of mutaton operators durng the search process. The BAP benchmark nstances that have been used by other researchers are used to assess the performance of the proposed algorthm. Our expermental results reveal that the proposed DE can obtan compettve results wth less computatonal tme compared to exstng algorthms for all tested problem nstances. Keywords: Dfferental evoluton, berth allocaton problem, meta-heurstcs, optmzaton. 1 Introducton Martme transportaton has experenced a tremendous growth of contaner usage over the last two decades [1], [2]. Port managers face great challenges n provdng effectve and effcent servces. The berth allocaton problem (BAP) s one of the man challenges confrontng port managers. Provdng an effcent soluton to the BAP plays an mportant role n mprovng port effectveness [2]. BAP seeks to assgn, for each vessel, a berth poston and servce tme on the selected berth. The goal s to mnmze the total watng tme of all vessels as far as possble [1]. BAP s an NP-hard problem [1]. Small nstances can be solved optmalty usng exact methods. However, they become mpractcal as the sze of the nstances n- Sprnger Internatonal Publshng Swtzerland 2015 H. Handa et al. (eds.), Proc. of the 18th Asa Pacfc Symp. on Intell. & Evol. Systems Vol. 2, Proceedngs n Adaptaton, Learnng and Optmzaton 2, DOI: / _7 77

2 78 N.R. Sabar, S.Y. Chong, and. Kendall creases [2]. As such, researchers have utlzed meta-heurstc algorthms to deal wth large-scale nstances as they can often provde good qualty solutons wthn realstc computatonal tmes. Examples of meta-heurstc algorthms beng utlzed for BAP nclude: tabu search [1], clusterng search [3] partcle swarm optmzaton [4] and hybrd column generaton approach [5]. In ths paper, we propose a dfferental evoluton (DE) algorthm for the BAP. DE s a nature-nspred populaton-based meta-heurstc that has been demonstrated to be effcent and effectve for many hard contnuous optmzaton problems. DE operates on a populaton of solutons and teratvely mproves them. In each teratve step or generaton, a new soluton s generated usng two varaton operators: mutaton and crossover. The mutaton operator n DE s consdered the prmary varaton operator and several mutaton operators have been ntroduced. However, t s not known n advance whch operator should be used as dfferent operators work well for dfferent problem nstances wth dfferent characterstcs and n dfferent stages of the search process [6], [7], [8], [9]. In ths paper, we propose an enhanced DE that utlzes a varaton operator wth multple mutaton operators for the BAP. We utlze game theory to provde a mechansm to control the selecton of mutaton operators throughout the search process of DE. The performance of the proposed algorthm has been assessed usng the exstng BAP benchmark nstances [1]. The expermental results reveal that the proposed algorthm can obtan compettve results wth less computatonal tme when compared wth exstng algorthms. 2 Problem Descrpton Berwrth and Mesel [2] has classfed BAP nto two types accordng to the berth type and the vessel s arrval tme. The berth type s categorzed as dscrete f the quay has been parttoned nto a set of berth sectons and contnuous f the quay s not parttoned. The vessel s arrval tme s categorzed as dynamc f the vessels can arrve at any tme durng the plannng horzon, and statc f all vessels have arrved at the port before the berth plannng begns. We focus on the dscrete dynamc BAP [2], [10]. For ths BAP, there are a set of berth sectons wth predefned lengths and a set of vessels. Each vessel has an arrval tme, prorty, vessel length and handlng tme [2]. Some of them can be allocated to any berths based on vessel lengths whle others can only be allocated to a subset of berths. The vessel handlng tme s dfferent from one berth to another. The overall goal s to allocate for each vessel a berth secton and servce tme (berthng tme) on the allocated berth whle respectng the followng constrants [1]: Each vessel s allocated to exactly one berth. There s no more than one vessel allocated to the same berth at the same tme (same servce tme). Each berth can handle at most one vessel at any gven tme.

3 A Hybrd Dfferental Evoluton Algorthm 79 The man role of the optmzaton algorthm s to mnmze the total watng tme of all vessels whch s calculated as follows (objectve functon) [1]: mn v n k m T k k a ++t where - n : number of vessels - m : number of berths - v : the prorty of vessel - k T : the bertng tme of a vessel at berth k. - a : the arrval tme of vessel. - k t : the handlng tme of vessel at berth k. k x j j n (1) - x j k : decson varable, x j k =1 f vessel j s servced by berth k after the vessel and x j k = 0 otherwse. 3 Proposed Methodology In ths secton, we frst present the basc DE algorthm that s followed by the proposed approach to enhance DE for the BAP problem. 3.1 Basc Dfferental Evoluton Algorthm The DE algorthm was proposed n [11] to deal wth contnuous optmzaton problems (real-valued ftness functons). It belongs to a class of nature-nspred, populaton-based meta-heurstc algorthms [12]. A general DE algorthm starts wth a populaton of solutons and then apples evolutonary operators of varaton (mutaton and crossover) and selecton to mprove the populaton of solutons teratvely over a certan number of generatons. For every soluton n the populaton, DE generates a new soluton usng the mutaton operator that randomly selects three dfferent solutons from the current populaton and combnes them accordng to a prescrbed operaton. The new generated soluton s then combned wth the parent soluton usng the crossover operator to generate an offsprng. The selecton step frst calculates the ftness of the offsprng and then replaces t wth the parent soluton f t has a better ftness. Ths process s repeated for a predefned number of generatons. Over the years, many DE varants that use dfferent mutaton or crossover operators have been proposed. A general DE scheme use the notaton DE/x/y/z, where x

4 80 N.R. Sabar, S.Y. Chong, and. Kendall represents the base soluton to be mutated, y defnes the number of dfferent solutons to be used to perturb x, and z denotes the crossover type, bnomal or exponental [11], [12]. A well-known DE varant s the DE/rand/1/bn, where DE s Dfferental Evoluton, rand means the solutons wll be randomly selected, 1 ndcates the number of pars of solutons and bn ndcates that bnomal crossover wll be used. The basc steps of the DE/rand/1/bn are as follows [11]: Step 1: Randomly generate a populaton of solutons, NP. Step 2: Calculate the ftness, f, of the populaton. Step 3: For each parent soluton ( x ) n the current populaton NP ( s the soluton ndex and s the current generaton) generate a new soluton ( m ) usng (2): m = x + F * ( x x ), j {1,..., n} (2), j 1, j 2, j 3, j where j represents the ndex of current decson varable, n s the maxmum number of decson varables n a gven problem nstance, F s the scalng factor (F [0, 1]) and x 1, x 2 and x 3 are three randomly chosen solutons from the current populaton where x1 x2 x3. Step 4: Apply the crossover operator to combne the soluton ( m ) generated by the mutaton operator wth the parent soluton ( x ) based on the crossover rate 1 CR (CR [0, 1]) n order to generate a new offsprng ( m + ) as follows (3): m, ( ) ( ) 1 j f Rand j CRor j = Rnd m +, j = (3) x, j f Rand ( j ) > CR and j Rnd ( ) j {1,..., n}, {1,..., NP } where Rand(j) s a random number (Rand(j) [0, 1]) selected for the j th decson varable, Rnd() s a random decson varable ndex (Rnd() {1,, n}). 1 Rnd ensure that m + gets at least one decson varable from m. 1 Step 5: Calculate the ftness of m + and compare t wth x. Replace x wth 1 m + 1 f m + ftness s better than x as follows (4): m ( ) ( ) 1 f f m f x + x = + 1 x f f ( m ) > f ( x ) {1,..., NP } (4)

5 A Hybrd Dfferental Evoluton Algorthm 81 Step 6: If the termnaton crteron s satsfed (the number of generatons), stop and return the best soluton. Otherwse, go to Step 3. 4 The Proposed Algorthm The varety of problem nstances havng dfferent characterstcs, or those wth complex structures (e.g., dfferent sectons havng dfferent structures), makes t challengng to know n advance the best varaton operator to use (.e., blackbox optmzaton) [12]. Each one has ts own strength and weakness and may work well for certan nstances or at a certan stages n the search process. Consequently, several DE frameworks that utlze a set of mutaton operators have been proposed [6], [7], [8], [9]. These frameworks seek to combne the strength of several mutaton operators n one framework that can effectvely solve the gven problem nstances [13], [14], [15]. In ths paper, we propose a DE algorthm that utlzes several mutaton operators to solve the BAP. The proposed DE makes use of game theory n the desgn of the selecton mechansm that chooses the operator to be used at the current search stage. In the followng, we frst descrbe the soluton representaton and the populaton generaton method, and later the utlzed mutaton operators and proposed selecton mechansm. 4.1 Soluton Representaton and the Populaton eneraton Method DE was orgnally proposed to solve contnuous optmzaton problems [12]. To deal wth combnatoral optmzaton problems such as BAP, a sutable soluton representaton or a decodng scheme s needed to convert the real numbers nto ntegers [12]. In BAP, each vessel has to be assgned to a berth secton and beng gven a servce tme on the selected berth. Ths mples that we need to deal wth the assgnment problem that s responsble for assgnng for each vessel a berth secton and the schedulng problem that assgns a servce tme for each vessel. In ths paper, we avod modfyng the DE mutaton operator by havng a decmal representaton of BAP solutons a 0.a 1 a 2...where the nteger part a 0 represents assgned berth sectons whle the fracton parts a 1 a 2... represent the order of ths vessel on ths berth [16]. Fgure 1 shows an example of BAP soluton representaton. Here, an nstance of BAP has 6 vessels (n=6) that are needed to be assgned to 3 berths. If we assgn numbers for the vessels from 1 to 6, the decson varables wll be (1, 2, 3, 4, 5, 6), as shown n the frst row of Table 1. Next, we generate for each decson varable a random number r (r [1, 3])), as shown n second row of Table 1. Table 1 can be decoded nto a BAP soluton as follows: vessel 1 s assgned to the second berth and s second n order on ths berth, vessel 2 s assgned to the frst berth and s frst n order on ths berth, vessel 3 s assgned to the frst berth and s second n order on ths berth, and so on. Next, on each berth we sort the assgned vessels n an ascendng order based on ther arrval tme. In ths paper, the ntal populaton of solutons of DE s randomly generated by assgnng each decson varable a random value between 1 and the maxmum number of vessels n a gven problem nstance. The generated solutons are assgned ftness values usng equaton (1).

6 82 N.R. Sabar, S.Y. Chong, and. Kendall Table 1. BAP soluton representaton Vessel ndex Decson varables Mutaton Operators and the Selecton Mechansm In ths paper, the proposed DE makes use of the followng mutaton operators [11], [12]: - M 1 : DE/rand/1/bn, m = x1 + F *( x2 x3) - M 2 : DE /best /1/bn, m = xbest + F *( x1 x2) - M 3 : DE /parent-to- best/1, m = x + F *( xbest x) + F *( x1 x2) - M 4 : DE /best /2/ bn, m = xbest + F *( x1 x2) + F *( x3 x4) where parent (x ) s the parent soluton to be perturbed, best ndcates the best soluton n the populaton and x 1, x 2, x 3 and x 4 are randomly selected solutons from the current populaton where x 1 x 2 x 3 x 4 x x best In ths paper, each parent soluton s assocated wth a set of mutaton operators and, at each generaton, one of them s selected to generate a new soluton. We employ a game theoretc concept to desgn the selecton mechansm of the mutaton operators for each parent. Each strategy s assocated wth a profle that keeps the hstory of the strategy performance. As n [17], we model the selecton mechansm n the context of a two-player game. The DE populaton of solutons s dvded nto two to play two-player game. Each soluton represents a player wth a set of mutaton operators (M 1, M 2, M 3 and M 4 ) representng strateges that can be played. Each strategy wll be assgned a payoff representng the mprovement obtaned by the selected strategy. Accordng to the strategy probablty dstrbuton, each player selects one strategy to play aganst another strategy. Based on the obtaned result, the player wll update the strategy profle and the payoff. In ths paper, the strategy profle keeps the accumulated payoff of each one and t s updated at every generaton. The payoff of each strategy s calculated as follows: let S [] be the array of the probablty of selectng the strategy, f p and f n represents the ftness values of the parent and generated solutons, NS represents the number of strateges. Then, f the applcaton of the -th strategy mproves the ftness value of the parent soluton, the payoff of the -th strategy s updated as follows: S[]=S[]+ where =(f p f n )/ ( f p + f n ), j {1,,NS} and j, S[j]=S[j]-( /(NS-1)). Otherwse (f the soluton cannot be mproved), S[]=S[]- ( *α) where α=current_enetraon/total_eneratons, j {1,,NS} and j, S[j]=S[j]+( *α/(ns-1)). Intally, the selecton probablty of each strategy s set to 1/NS.

7 A Hybrd Dfferental Evoluton Algorthm 83 5 Expermental Setup The BAP benchmark nstances that have been ntroduced n [1] are used to valdate the performance of the proposed algorthm. The benchmark nvolves 30 dfferent nstances (denoted as 1 to 30); each nstance has 30 vessels and 13 berths [1]. In all nstances, the vessel characterstcs such as length and the arrval tme as well as berth lengths are dfferent. Table 2 shows the parameter settngs of proposed algorthm. These settngs were determned based on prelmnary experments. In ths paper, we executed the proposed algorthm 31 tmes for each nstance usng dfferent random seeds. Table 2. The Parameter Settngs # Parameter Value 1 No. Of generatons Populaton sze, NP 20 3 Scalng Factor, F Crossover Rate, CR The Computatonal Results We have carred out two types of experments. The goal of the frst one s to evaluate the mpact of the proposed mult-mutaton operators on the performance of DE n solvng BAP through a comparson wth a standard, baselne DE (DE/rand/1/bn). The goal of second experment s to compare the results of the proposed algorthm aganst the state of the art algorthms. 6.1 The Computatonal Comparsons of DE wth and wthout the ame Theory Concept In ths secton, we compare the computatonal results of DE wth and wthout the game theory concept (denoted as DET and DE, respectvely) usng the same parameter settngs, stoppng condton and computer resources. Both algorthms (DET and DE) are executed for 31 ndependent runs and the results are compared usng the Wlcoxon statstcal test wth a sgnfcance level of The p-value of DET aganst DE for all nstances s presented n Table 3. In ths table, + ndcates DET s statstcally better than DE (p-value < 0.05), - ndcates DE s statstcally better than DET (p-value > 0.05), and = ndcates both DET and DE have the same performance (p-value = 0.05). As Table 3 reflects, DET s statstcally better than DE on 21 nstances and preforms the same as DE on 2 out of 30 tested nstances. Ths table also reveals that on 7 out of 30 tested nstances, DET s not statstcally better than DE. Although the results show that DET s not statstcally better than DE on all tested nstances, the overall fndng justfes the beneft of ntegratng the game theory concept wth DE algorthm. Indeed, the use of the game theory concept can effectvely enhance the performance of DE to obtan very good results for all tested nstances.

8 84 N.R. Sabar, S.Y. Chong, and. Kendall Table 3. The p-value of DET compared to DE DET vs. DE Instance p-value = 06 = The Computatonal Comparsons of DET wth State of the Art Algorthms In ths secton, we compare the computatonal results of DET wth the current state of the art algorthms. The algorthms that we compare aganst are: - eneralzed set partton programmng (SPP) [18]. - Tabu search (TS) algorthm [1]. - Column generaton (C) algorthm [5]. - Clusterng search (CS) [3]. - Partcle swarm optmzaton (PSO) [4].

9 A Hybrd Dfferental Evoluton Algorthm 85 Table 4 gves the results of DET over 31 runs as well as the compared algorthms. For each nstance, we present the best obtaned results (best objectve value) and the computatonal tme (seconds) obtaned by DET and the compared algorthms. In Table 4, the thrd column (Opt.) ndcates the optmal value for each nstance [18], the last row represents the average overall nstances (Avg.) and boldfont ndcates the best obtaned results. Table 4. The results of DET compared to the state of the art methods Inst. DET SPP TS C CS PSO Best Tme Opt. Tme Best Best Tme Best Tme Best Tme Avg As can be seen from Table 4, DET obtaned the optmal values for all tested nstances. In partcular, DET best results are the same as those produced by the SPP. Wth respect to ndvdual comparsons, DET best results are the same as CS, PSO, TS and C on 30, and 24 out of 30 tested nstances, respectvely. DET obtaned better results than TS on 18 and C on 4 nstances. In addton, the average result of all nstances (last row n Table 4) of DET s better (or the same) than the compared algorthms. As for the computatonal tme comparsons, Table 4 shows that, on all tested nstances, the computatonal tme of DET s lower than SPP, CS, PSO, TS and C. The overall results demonstrate that DET s an effectve and effcent algorthm for the BAP as t obtaned very good results for all tested nstances wthn a small computatonal tme when compared to prevously reported algorthms.

10 86 N.R. Sabar, S.Y. Chong, and. Kendall 7 Concluson In ths paper, we have presented a DE algorthm to solve the BAP. DE s a populaton based algorthm that seeks to mprove the populaton of solutons through the use of mutaton operator(s), crossover operator and selecton rule. To further enhance the performance of the DE, we coupled t wth a several mutaton operators n order to combne strength of dfferent operators n one framework. ame theory s used to control the selecton of whch mutaton operator should be used at any decson pont. The computatonal results are carred out usng the exstng BAP benchmark nstances. The obtaned results reveal that the proposed algorthm obtaned very good results when compared to DE wthout game theory as well as the state of the art algorthms. In addton, the computatonal tme of proposed algorthm s lower than the compared algorthms, ndcatng that proposed algorthm s an effectve algorthm for the berth allocaton benchmark nstances. References 1. Cordeau, J.-F., Laporte,., Legato, P., Mocca, L.: Models and tabu search heurstcs for the berth-allocaton problem. Transportaton Scence 39(4), (2005) 2. Berwrth, C., Mesel, F.: A survey of berth allocaton and quay crane schedulng problems n contaner termnals. European Journal of Operatonal Research 202(3), (2010) 3. de Olvera, R.M., Maur,.R., Noguera Lorena, L.A.: Clusterng Search for the Berth Allocaton Problem. Expert Systems wth Applcatons 39(5), (2012) 4. Tng, C.-J., Wu, K.-C., Chou, H.: Partcle swarm optmzaton algorthm for the berth allocaton problem. Expert Systems wth Applcatons 41(4), (2014) 5. Maur,.R., Olvera, A.C.M., Lorena, L.A.N.: A hybrd column generaton approach for the berth allocaton problem. In: van Hemert, J., Cotta, C. (eds.) EvoCOP LNCS, vol. 4972, pp Sprnger, Hedelberg (2008) 6. Mallpedd, R., Suganthan, P.N., Pan, Q.-K., Tasgetren, M.F.: Dfferental evoluton algorthm wth ensemble of parameters and mutaton strateges. Appled Soft Computng 11(2), (2011) 7. Qn, A.K., Suganthan, P.N.: Self-adaptve dfferental evoluton algorthm for numercal optmzaton. In: The 2005 IEEE Congress on Evolutonary Computaton, pp IEEE (2005) 8. Brest, J., Boškovć, B., rener, S., Žumer, V., Maučec, M.S.: Performance comparson of self-adaptve and adaptve dfferental evoluton algorthms. Soft Computng 11(7), (2007) 9. Zhang, J., Sanderson, A.C.: JADE: adaptve dfferental evoluton wth optonal external archve. IEEE Transactons on Evolutonary Computaton 13(5), (2009) 10. Sabar, N.R., Kendall,., Ayob, M.: An Exponental Monte-Carlo Local Search Algorthm for the Berth Allocaton Problem. In: 10th Internatonal Conference on the Practce and Theory of Automated Tmetablng (PATAT 2010), York, UK, August 26-29, 2014, pp (2014) 11. Storn, R., Prce, K.: Dfferental evoluton a smple and effcent heurstc for global optmzaton over contnuous spaces. Journal of lobal Optmzaton 11(4), (1997)

11 A Hybrd Dfferental Evoluton Algorthm Das, S., Suganthan, P.N.: Dfferental evoluton: A survey of the state-of-the-art. IEEE Transactons on Evolutonary Computaton 15(1), 4 31 (2011) 13. Sabar, N.R., Ayob, M., Kendall,., Rong, Q.: rammatcal Evoluton Hyper-Heurstc for Combnatoral Optmzaton Problems. IEEE Transactons on Evolutonary Computaton 17(6), (2013), do: /tevc Sabar, N.R., Ayob, M., Kendall,., Qu, R.: A Dynamc Multarmed Bandt-ene Expresson Programmng Hyper-Heurstc for Combnatoral Optmzaton Problems. IEEE Transactons on Cybernetcs PP(99), 1 (2014), do: /tcyb Sabar, N.R., Ayob, M., Kendall,., Qu, R.: The Automatc Desgn of Hyper-heurstc Framework wth ene Expresson Programmng for Combnatoral Optmzaton problems. IEEE Transactons on Evolutonary Computaton PP(99), 1 (2014), do: /tevc Sabar, N.R., Kendall,.: Arcraft Landng Problem usng Hybrd Dfferental Evoluton and Smple Descent Algorthm. Paper presented at the 2014 IEEE Congress on Evolutonary Computaton (CEC 2014), pp (2014)