Neigborhood Selection in Variable Neighborhood Search

MIC 2011 : The IX Metaheur istics Intern ational C onfere nce S2-15–1 Neigborhood Selection in V ariable Neighborhood Sear ch M. J. Geiger 1 , M. Sev a ux 1 , 2 , Stefa n V oß 3 1 Helmut Schmidt Univ ersity , Logistics Manag ement Department Holstenhofw eg 85, 22043 Hamb u rg, Germany m.j.geiger @hsu-hh.d e 2 Universit ´ e de Bretagne-Sud, Lab-STICC – Centre de Recherche 2 rue de St Maud ´ e, F -5632 1 Lorient, France marc.sev aux@univ-ubs.fr 3 University of Hamburg, Institute of Information Systems (W irtschaftsinform atik), V on-Melle-Park 5, 20146 Hamburg, Germany stefan.voss@ uni-ham burg.de 1 Introd uction V ariable neighbo rhood sear ch (VN S) [4, 5] is a m etaheu ristic fo r solving optimization problems based o n a simple principl e: systemat ic ch anges of neighbor hoods within the s earch, both in the descent to local minima and in the escape from the v alle ys which contain them. VNS applicat ions hav e been numerous and succes sful. Many extensio ns hav e been made, mainly to be able to solve lar ge problem instances. Nev ertheless, a m ain driv er behind VN S is to keep the simplicity of the basic scheme. V ariable neigh- borho od descend (V ND) belongs to the family of methods within VNS. The idea behind VN D is to sys- tematical ly switch between diffe rent neighborho ods where a local search is compreh ensi vely performed within one neighborh ood until no further impro vemen ts are possible. Give n the found local optimum, VND continues with a local search within the next neighborho od. If an impro ved solution is found one may resort to the first neigh borho od; otherwise the search continues with the next neighb orhoo d, etc. Designin g these neighbor hoods and applyin g the m in a meaningful f ashion is not an e asy task. More- ov er , inco rporat ing thes e neighborh oods, e.g., into V NS may still need a cons iderabl e deg ree of ingenuity and may be con fronted w ith the follo wing research quest ions (among others): • Which neig hborho ods can be designed ? • Ho w should they be explo red and how ef ficient are they? • Ho w structurally differe nt are they? • In which orde r should they be applie d? In this paper we attempt to in ve stigate especially the latter concern. Assume that we are giv en an optimiza tion proble m that is intend ed to be solve d by apply ing the VNS scheme, how many and which types of neighborh oods should be in vestiga ted and what could be appropri ate selectio n criteria to apply these neighbo rhoods . More specifically , d oes it pay to “look ahead” (see , e.g., [6] in the contex t of VNS and GRASP ) when attempting to switch from one neighborho od to another? Is it reasonable to apply “nest ed neigh borho od structures” (like first 2-optimal exc hanges , then 3-optimal exchange s, then 4-opti mal e tc.) or should neighborhoo d structure s be considera bly dif ferent fro m each oth er? Often dif ferent neighborho ods are proposed allowing for idea s regarding seque ntial and nested changes. W e ca nnot pro vide a co mprehens i ve surv ey on t he VNS in this e xtended abstra ct. W e emphasi ze just ver y fe w earlier works on VNS in vestigat ing some of the question s raised. For instanc e, in [1 ] VNS is hybridiz ed with a gen etic algo rithm (GA). Chromosomes in the GA repres ent the possibl e choices of ne ighbor hoods to be appli ed. H o we ver , as the authors state, “the order - ing of neighbo rhood s is unimportan t since VNS cycles through all neighb orhood s. ” On the other hand, when hybridi zing the VNS with the pilot method as it is performed in [6 ] it actually seems fav orable to put some effort in an appro priate choice of the next neighborh ood. Problems consider ed are ex am timetabli ng in the first case and telecommunica tions network d esign in the latter . These r esults ar e in line with the obser v ations in [7] for apply ing V ND on a dif ferent network design problem. Udine, Italy , July 25–28 , 2011 S2-15–2 MIC 2011: The IX Metaheurist ics Internati onal Conferenc e T o summarize, dif ferent s ources come to differe nt conclusion s making it ev en more important to in ves tigate these topics further . While our research quest ions are of generic natur e we attempt to in- ves tigate them and provide answers w ith respec t to the single machi ne total w eighte d tar diness pr oblem (SMTWTP) as a specific combinato rial optimizatio n problem. Though this se tting might no t necess arily lead to ge neraliz able results (as is the case with the re ferenc es cited abo ve) the y se em to allo w additi onal insigh ts into the VN S method in itself. 2 VNS f or the Single Machine T o tal W eighted T ardiness Pr oblem The S MTWTP or 1 | | P w j T j is a well-kno wn mach ine scheduli ng problem. In the S MTWTP , a set of n jobs with w eights indicatin g their relati ve importance needs to be proces sed on a single machine while minimizing the total weighted tardines s. For a recent paper on the SMTWTP plus some related referen ces see, e.g., [3]. W ith respect to our above mentioned research questi ons, the SMT WTP may be seen as a represen - tati ve problem with the follo wing characteris tics: easy to represent by some straightf orward encoding, hard to solv e, an exist ing testbed of problem instances with kno wn optimal solu tions is readily av ailable for deta iled analysis. Neighborhoods f or the SMTW TP . The set of ne ighbor hoods to be ex plored should be importan t enoug h to get some insights regardi ng our purpose. W e ha ve decided to buil d a set of neighborh oods compose d by the cla ssical ones and by some structurally differe nt neighborh oods. The in ves tigate d neighbo rhood s includ e the follo wing: APEX it is th e most cla ssical neighbo rhood for permuta tion encoding s. T wo ad jacent jobs are e x- chang ed in the seque nce. For a solu tion, the size of this neighbo rhood is n − 1 . BR4 cons ists in taking a block of four consecuti ve jobs and in ver se its internal orient ation. The size of this neigh borhoo d is n − 3 . BR5 is ide ntical to BR4 but with a blo ck of fiv e jobs. The size of this neighbo rhood is n − 4 . BR6 is ide ntical to BR4 but with a blo ck of six job s. The size of this neighb orhood is n − 5 . EX \ APEX is the gen eral pair wise int erchan ge where th e APEX is e xcluded. T wo n on-adj acent jobs a re exc hanged . The size of this neighborho od is n ( n − 3) + 2 . FSH \ APEX tak es a job and moves it to a position further in the sequence , resulting in a forward shift for the jobs in between these two position s. APEX is excl uded from this mov e. The size of this neighb orhoo d is P n − 2 i =1 i = ( n − 2)( n − 1) / 2 . BSH \ APEX wor ks as FSH \ APEX b ut the jobs are shifted backward . T he size is identical to FSH \ APEX. Experimental Design. Since we want to obser ve the beha vior of the search when selec ting the dif ferent neighb orhoo ds, we pr opose three dif ferent approac hes. T o simplify the exp eriments and not b e too depen dent of randomness, instead of using a VNS, we first use a VND sch eme. In the ex periments , we will compare the three follo wing algori thms: VND-R When a local optimum is reached, the next neigh borho od is selected randomly among possible alterna ti ves. The search terminate s when all the neighb orhood s cannot produc e a better solutio n than the incumbe nt. VND-F T his is the classical versi on of VND. In our experiments , we choose the order in which neigh- borho ods are introd uced abov e. VND-A T his is an adapti ve version of VND. Every time a ne w neigh borho od has to be selected, an in ves tigatio n of all neighborhoo ds is run for a short number of iteratio ns (say , 100) and the neigh- borho od produci ng the be st so lution at this t ime is sel ected for the descent method. This neighb or - hood is used until no more improv ements are obtaine d. The in ves tigatio n phase is run again, etc. The stopp ing condi tion is the same as befo re. T o make a fair compariso n, for all appro aches we count the number of ev aluations made during the search es and we sh all u se gr aphs indica ting the quality of the be st so lution v s. the number of ev aluations. T o consi der neste d neighborho ods, the abov e excl usion may be abo lished , too. Udine, Italy , July 25–28, 2011 MIC 2011 : The IX Metaheur istics Intern ational C onfere nce S2-15–3 3 Pr eliminary r esults Experiment s are condu cted on the c lassica l 100 job ins tances from [ 2]. These ins tances ha ve some adv antages: best kno wn solutions are stable (not improv ed for a long time), the number of instan ces is not t oo la r ge (125 ) and these instanc es still seem difficu lt to solve (des pite the ir relati vely sma ll siz e with 100 job s). In Figure 1 we pr ovid e as an ex ample the re sults of the first r un on the firs t of the in ves tigated set of problem instances. In this case we find that VND-R take s longer to stop than VND-F , and that tak es more time tha n VND-A. That is, we may d educe a pos iti ve aspect of the adapti ve vers ion of VND. W e can also see that if we run VND for less than 750 ev aluation s, VND-R giv es best results, and less than 834 e v aluations giv e priorit y to VN D-F . O f course , 834 ev aluation s are not enough to reach good qualit y and V ND-A sho ws its strength here. 5 10 15 20 25 30 35 40 45 50 55 60 0 5 10 15 20 25 30 35 40 45 50 Function value (x1000) Number of evaluations (x1000) Random Fixed Adaptative End of VND-A End of VND-F End of VND-R VND-A improves VND-R VND-A improves VND-F Figure 1: Example run for the first instance of [2] This ideal si tuatio n was unf ortuna tely not observ ed as a g eneral beh a vior and we re port consi derably more details in the final version of the paper together with related conclusio ns and proposals for related adapti vity in VNS and VND with the final conc lusion that adapti vity actually pays. Refer ences [1] E. K. B urk e, A.J. Eckersl ey , B. McCollum, S . Petro vic, and R. Qu. Hybrid varia ble neighbo urhoo d approa ches to uni versity exam timetablin g. Eur opea n Journ al of Oper ationa l Resear ch , 206:46 –53, 2010. [2] H. A.J. Crauwels , C.N. Potts, and L.N. V an W assenho ve. Local searc h heuri stics for the single machine total weighted tardin ess sched uling problem. INFORMS J ourna l of Computing , 10:341– 350, 1998. [3] M.J. Geiger . On heuristic search for the single m achine total weighted tardin ess problem - some theore tical insigh ts and th eir empi rical v erificatio n. Eur opean J ournal of Operati onal Resear ch , 207:1 235–1 243, 2010. [4] P . Hansen and N. Mladeno vi ´ c. An introduction to v ariable neighborho od search. In S. V oß, S. Martello , I.H. Osman, and C. Roucairol, editors, Meta-Heuristic s: Advance s and T ren ds in Local Sear ch P arad igms for Optimization , pages 433–45 8. Kluwer , Boston, 1999. [5] P . Han sen and N. Mladeno vi ´ c. V ariab le neighbo rhood search. In F . Glo ver and G. A. K ochenber ger , editor s, Handbo ok of Metaheu ristic s , pages 145–184 . Kluwer , Boston, 2003. [6] H. H ¨ oller , B. Melian, and S . V oß. Applying the pilot method to improv e V NS and GRASP meta- heuris tics for the design of SDH/WDM networks. Eur opean J ournal of Operation al Resear ch , 191:6 91–70 4, 2008. [7] B. Hu and G .R. Raidl. V ariable neighbor hood descent with self-adapti ve neighborh ood-o rdering. In Pr oceedings of the 7th EU/Meeting on Adaptive , Self-Adaptiv e, and Multi-Lev el Metaheu ristics . Malaga, Spain, 2006 . Udine, Italy , July 25–28 , 2011