Success of Heuristics and the Solution Space Representation

Transcription

1 Magyar Kutatók 7. Nemzetköz Szmpózuma 7 th Internatonal Symposum of Hungaran Researchers on Computatonal Intellgence Success of Heurstcs and the Soluton Space Representaton Sándor Csszár Department of Mcroelectroncs and Technology, Kandó Kálmán Faculty of Electrcal Engneerng, Budapest Tech Tavaszmező út 7, H-084 Budapest, Hungary csszar.sandor@kvk.bmf.hu Abstract: In the study a developed soluton for VRP (Vehcle Routng Problem) and ts algorthms were analysed n order to fnd explanatons for ther success and evaluate ther results. Another exctng queston at the combnatoral optmzaton s the soluton space whch s often referred n the lterature despte we do not have tools for ts representaton, although the dstncton of solutons seems to be mportant at the Evolutonary technques and at the generalzed model of Adaptve Memory Programmng. Keywords: combnatoral optmzaton, heurstcs, vehcle routng problem Introducton A two-phase metaheurstc wth guded route elmnaton was developed by the author [] for the Vehcle Routng Problem wth Tme Wndows (VRP TW) to solve a mddle sze ( n = 00 ) combnatoral optmzaton problem. It s known from experence and from the lterature that heurstcs are very successful despte ther performance s not ustfed only some convergence theorems are proved [3]. Several questons often arse regardng the performance of the heurstcs, optmalty of the found soluton, navgaton n the soluton space, and how more effcent methods can be found. It s really dffcult to gve general answers for these questons (perhaps mpossble for one or the other). The maor part of the presented results comes from computaton and from analyss of a specfc problem. The purpose of ths artcle s to better understand and evaluate the performance of the developed algorthms n order to solve large scale problems -where the number of customers s about 000 and to put these questons nto focus. 383

2 S. Csszár Success of Heurstcs and the Soluton Space Representaton Regardng the topc of ths artcle (detaled above) only a short VRP defnton s gven. Let G = {V, E} be a graph and V = {v 0, v,..., v n } a set of vertces representng the customers around a depot, vertex v 0 denotes the depot. A route s startng from and enterng the depot: {v 0,..., v, v,..., v 0 }. The cost of a route s: C t = d, where and are consecutve customers on the route and d, E. The obectves of the soluton are to determne the lowest number of routes (N v ) and the lowest cost or total travel dstance, C = (s) Ct, where s s the actual soluton, provded that each customer can be vsted only once and should satsfy all the constrants (tme wndow, capacty). 2 Investgaton of Heurstcs Based on the Developed Soluton 2. Evaluaton of the Found Best Solutons The computatonal program developed for the cted study [] and the Solomon benchmark set [4] ncludng 56 problem nstances was used for the nvestgaton. The benchmark set was developed by M. M. Slomon to compare the behavour of dfferent heurstcs on dverse problems. It seems to be useful and nformng to compare the found best result and the estmated lower cost lmt of the gven problem nstance. The lower cost lmt was computed the followng way: The number of arcs of the graph s n ( n ) = 9900 (drected graph). The shortest 2 of { v 0 v } {,2 00} arcs were selected because n case of Nv Nv routes there are as many routes startng from and endng at the depot. The further number of arcs ( 00 Nv ) were selected from the remander, smlarly the shortest ones. At ths latest selecton the dentcal values were deleted because only one of them can be appled n a feasble soluton (f v arc s used then v s excluded). Consderng the large number of problem nstances (56) and the dversve problem types (random: R00 and R200, cluster: C00 and C200, and semclustered: RC00 and RC200) the found solutons by heurstcs are qute close to the estmated lower cost lmt (the reference s the random arc selecton). The results are llustrated by problem type, because of the slght dfference between the ndvdual problems. In Fgure the random arc selecton, the lower cost lmt and the found best result by heurstc are compared. It must be noted that at the random selecton the feasblty of the soluton was not consdered whle at the cost lmt calculaton a statc feasblty was consdered. It means a route 384

3 Magyar Kutatók 7. Nemzetköz Szmpózuma 7 th Internatonal Symposum of Hungaran Researchers on Computatonal Intellgence { v v } s feasble f Te + ts + d + ts Tl () where ts s the servce tme at node, d s the dstance between and, Tl s the latest start of servce at node, Te s the earlest start of servce at node. Under route constructon the condton determned by Equaton () s usually not suffcent because only a part of the tme wndow s avalable for travel. Table n the Appendx gves data for all the sx problem types Random edge selecton Estmated cost lmt Best found C00 Fgure Heurstc performance (clustered problem type) The nvestgated heurstcs gve qute good results despte they use a so called nexact logc. Explanatons for performance are looked for n the next secton n case of a specfc problem. The followng queston s nvestgated n the next secton: s t consequental to get good results wth heurstcs? 2.2 The Nearest Neghbour Example on the Travellng Salesman Problem The Travellng Salesman Problem (TSP) s a specfc case of the VRP where only one route s allowed, consequently the number of soluton s: n!. Let s examne ths problem wth the nearest neghbour heurstc (one of the smplest route constructon). Let n = 9 (number of customers) be n the example (the arc lengths are not essental at ths pont). Apply the nearest neghbour heurstc for the TSP. Fgure 2 shows the route constructon. The numbers n the matrx ndcate the steps of the route constructon procedure. The numbers determne the lowest value wthn the elgble arcs n each row (the strategy of the nearest neghbour heurstc). 385

4 S. Csszár Success of Heurstcs and the Soluton Space Representaton Suppose that an arc of ( v v ) s selected. Two thngs can be establshed: In every step of the procedure the mnmum value of the gven row s selected then all the remanng arcs of ths row ( v s ) and the relevant column ( v r ) fall out of the further selecton ( r, s P, where P s the set of ndexes of arcs ndcated by x and O ). 2 If an arc ( v v ) s selected as a mnmum value, or travel dstance n a certan row- then obvously ( v v ) can not be selected. From the establshments above an nverse-symmetrc feature of the matrx s deduced (n the symmetrc postons of an x or a number, an O can be found) and the x and numbers n the matrx covers all the arcs of the graph x x x x x x x x 0 O O O O O O O O O x O x 5 O x x O O x x x x O x x 3 O x O O O O O 8 O O x O O x O 6 x O O x x 2 x x x x x O x O O 7 O O x O O 9 O O O O O O O O x 4 O x x O x x No of elgble arcs No of (x) S v9 S v8 S v7 S v6 S v5 S v4 S v3 S v2 S v S v0 46 No of (O) S h0 S h S h2 S h3 S h4 S h5 S h6 S h7 S h8 S h9 Fgure 2 TSP wth nearest neghbour route constructon The problem s the followng: the last arc (9) s determned by the prevous procedure, addtonally the last node determnes the route ( v 0 ) to the startng pont (there are two routes to the depot). Let Sh be the sum of the elgble arcs ( d ) of row, Sv that of the of the column. The performance of ths smple heurstc can be estmated f x arcs are put n order of magntude. Rearrange the rows -and the ndexes- of the matrx accordng to the decreasng number of x : S = d + d, where d d, {, 9} 6 (2) h ( 6)

5 Magyar Kutatók 7. Nemzetköz Szmpózuma 7 th Internatonal Symposum of Hungaran Researchers on Computatonal Intellgence Generally: S h = dmn ( ) + d (3) P Calculate the average arc: a = n( n + ) n n d = 0 = 0 ( ) (4) Let s handle the last arc ( d l ) dfferently. It s known that the numbered arcs ( d ) have the lowest value wthn the elgble elements. Examne the nearest neghbour heurstc, consderng the arcs sgned by x cover all the arcs of the graph: S k a = k d + k Δˆ k {, n} d Δˆ 0 (5) h = mn( ) mn( ) The cost of the soluton s: n n mn ( ) l l = = = = n C = d + d = a Δˆ + d = na Δˆ + a + Δl (6) C = ( n + ) a 0.5 n( n + ) Δ + Δ Δ 0 (7) l Δˆ and Δ are proportonal wth the dfferences of the arcs. If the length of the arcs are dentcal ( d = const ) then the cost of the soluton s ( n + ) a, but the found soluton s optmal. In any other case the optmal soluton s not guaranteed. The hgher Δ (or Δˆ ) s the closer the found soluton by the (nearest neghbour) heurstc to the estmated cost lmt. Ths condton exsts at the cluster type problems (C00, C200) where many close and far nodes are avalable n large number. Although these results were produced by much more sophstcated algorthms but the best numbers of routes at these problems can be easly acheved by ntal route constructon (Fgure 3). n 3 Soluton Space Representaton At practcal problems ( n > 50 ) the revealed part of the soluton space s very lmted. If we take the TSP example, n case of n = 00, t has 00! 9,33 exp(57) solutons. Usual computer programs wth 3000 teratons usng three operators and Global Best Strategy (examnng the whole 8 neghbourhood and selectng the best one) nvestgates maxmum 0 solutons, that s a neglgble part of the soluton space. Fortunately the heurstc solutons accordng to the experences converge very fast to the promsng zone and there s no need to search dsnterest regons. 387

6 S. Csszár Success of Heurstcs and the Soluton Space Representaton Presently we are not able to purposvely navgate nto the desred zone, but t would be mportant to know how far the dfferent solutons are from each other. The appled search processes don t provde any knowledge about t. It comes from the neghbourhood graph defnton [2] that wth the avalable operator set sometmes there s no way from one soluton to the other although these solutons are qute close to each other. An attempt was made n the remanng part of the artcle to represent the dstance of dfferent solutons by ther uncommon parts (uncommon arcs). The ntal solutons of [] were used for the exercse. The purpose of ths leads to the subect of Adaptve Memory Programmng (AMP). AMP s a syntheszed methodology. The populaton of genetc algorthms, the pheromone tral of the ant colony systems or the short- md - or long term memores of the tabu search play the same role. The basc prncple of AMP s to combne components of good solutons n order to construct other, hopefully better solutons. If the new soluton s better then the worst one stored n the memory t s updated wth the components of the actual one. [5]. In the adaptve memory a knd of crossover s made, consequently the requrement of the soluton (whch s nserted nto the memory) s not only the qualty but to have certan dfferences n order to brng new genes nto the next generaton of the solutons. The dfference between two solutons s supposed to show the dfferences n genes. Let 0 D be the dstance between two solutons, E, E E the set of arcs of solutons and, then: D card( E E ) = D = (8) mn{ card( E ), card( E )} The cardnalty of arcs n the dfferent solutons s usually the same, VRP s an excepton f the number of routes s dfferent. The mnmum cardnalty s suggested (the range of the receved number s wder). If we want to represent more then two solutons then a matrx should be used. In the Appendx a couple of examples are gven. In case of several solutons the common part of them s compound, but only the common pars are calculated. The calculaton of D s qute smple, the common part s a scalar product of the relevant matrxes. It s enough to store only those arcs whch are n the actual soluton (nstead of 0000 Bytes at VRP TW, n = 00 only ~20 arcs, 240 Bytes). At the AMP about 0 solutons are stored n the adaptve memory, occupyng 2400 Bytes for ths purpose. Tables 2 and 3 show examples. R203 supposed to have an extent, dverse soluton space because wth a few exceptons ( D 32, D4 ) for most of the other pars D > 0. 7 relaton s vald. C0 seems to have a narrow and deep soluton space. It s n accordance wth the dffcultes of computaton and the 388

7 Magyar Kutatók 7. Nemzetköz Szmpózuma 7 th Internatonal Symposum of Hungaran Researchers on Computatonal Intellgence establshments n ths artcle. Comng back to the purpose of dstance of solutons beyond the vsualzaton- t can be used as an ndcator at the refreshment of adaptve memory. Although the AMP algorthm ensures a certan dversfcaton, f contnuously very close solutons are added to the memory t does not help revealng new soluton space. Conclusons Ths study summarses thoughts about the success of heurstcs, analyse and explan the receved results, rase a few questons related to the soluton space, ntroduces dstance of solutons and prepare an Adaptve Memory Programmng study wth n = 000 customers. Acknowledgement The author wshes to thank the Department of Mcroelectroncs and Technology, Budapest Tech, Hungary frst of all Dr. Péter Turmeze, Drector of Kandó Kálmán Faculty of Electrcal Engneerng for ther support. References [] S. Csszár: Optmzaton Approaches for Logstc Problems -Vehcle Routng Problem wth Tme Wndows, PhD Dssertaton, Budapest, Hungary, submtted: August 2006 [2] I. Futó: Mesterséges Intellgenca (Aula Kadó), 999, pp [3] B. Haek: Coolng Schedules for Optmal Annealng, Mathematcs of Operatons Research 3, 200, pp [4] M. M. Solomon: Algorthms for the Vehcle Routng and Schedulng Problems wth Tme Wndows Constrants, Operaton Research 35, 987, pp [5] É. D. Tallard, L. M. Gambadella, M. Gendreau, J. Y. Potvn: Adaptve Memory Programmng: A Unfed Wew of Metaheurstcs, European Journal of Operaton Research 35, 200, pp

8 S. Csszár Success of Heurstcs and the Soluton Space Representaton Appendx Random edge selecton Estmated cost lmt Best found by heurstc R R C C RC RC Table Heurstc performance (by problem type) R Table 2 Soluton space representaton (R203) 390

9 Magyar Kutatók 7. Nemzetköz Szmpózuma 7 th Internatonal Symposum of Hungaran Researchers on Computatonal Intellgence C Table 2 Soluton space representaton (C0) Fgure 2 Computatonal Result of C00 Problem 39