JOURNAL OF COMPUTERS, VOL. 3, NO. 8, AUGUST 2008 77 Selfsh Constrant Satsfacton Genetc Algorthm for Plannng a Long-dstance Transportaton Network Takash Onoyama and Takuya Maekawa Htach Software Engneerng Co., Ltd., Tokyo, Japan E-Mal: {onoyama, taku}@htachsoft.jp Yoshtaka Sakura and Setsuo Tsuruta Department of Informaton Envronment, Tokyo Denk Unversty, Inza, Japan E-Mal: {ysakura, tsuruta}@se.denda.ac.jp Norhsa Komoda Graduate School of Informaton Scence and Technology, Osaka Unversty, Suta, Japan E-Mal: komoda@st.osaka-u.ac.jp Abstract To buld a cooperatve logstcs network coverng multple enterprses, a plannng method that can buld a long-dstance transportaton network s requred. Many strct constrants are mposed on ths type of problem. To solve ths problem effcently, a selfsh-constrant-satsfacton genetc algorthm (GA) s proposed. In ths type GA, each gene of an ndvdual satsfes only ts constrants selfshly, dsregardng constrants of other genes even n the same ndvdual. Further, to some extent, even ndvdual that volates constrants can survve over several generatons and has the chance of reparaton. Moreover, a constrant pre-checkng and dynamc penalty control methods are also appled to mprove convergence of GA. Our expermental result shows that the proposed method can obtan an accurate soluton n a practcal response tme. Index Term genetc algorthm, logstcs network, cooperatve logstcs, vehcle routng problem I. INTRODUCTION Logstcs cooperaton coverng multple enterprses s an effectve method for mprovng logstcs effcency of parts procurement []. However, ths cooperaton forms a cooperatve logstcs network consstng of several mutual sub-networks such as parts-collecton networks coverng parts supplers and depots (dstrbuton centers) and a long-dstance transportaton network coverng depots and factores. Thus, constructon of ths type of cooperatve logstcs network s therefore a very complcated task. Accordngly, usually ths task s dvded nto several phases [2, 3]. We already proposed a two-phased ( constructon phase and schedulng phase ) plannng method for optmzng a cooperatve logstcs network [4]. In the constructon phase, whch corresponds to strategc or tactcal level plannng, we make a parts-procurement plan correspondng to a producton plan and assgn resources such as drvers and vehcles to sub-networks [5, 6]. Next, n the schedulng phase, an actual transportaton schedule to evaluate the operablty of the network and to evaluate accurate logstcs cost based on the number of used vehcles s made. A smple mathematcal optmal plan s usually not readly accepted because of conflcts that arse n the dfferent enterprses. A human expert must thus evaluate the plan from many aspects and coordnate t wth consderaton of many non-systemzed condtons. To buld logstcs plans for the schedulng phase, soluton methods for schedulng a long-dstance transportaton network and for depot-centered parts-collecton networks whch provde at least human-expert-level accuracy wthn nteractve response tme are requred. We already proposed a VRP (Vehcle Routng Problem) [7, 8] soluton method for parts-collecton networks [9]. In ths paper, we propose a schedulng method for a long-dstance transportaton n the cooperatve logstcs network. A long-dstance transportaton network s not only drectly connected to the producton schedules of factores but also connected to many parts-collecton sub-networks. Moreover, ths network contans many transportaton ponts (depots and factores) dspersed all over the naton. And on the network, a vehcle dstrbutes and collects parts at the same tme. Strcter constrants are thus mposed on ths type of schedulng problem than those on a normal VRP. As a method for solvng VRP, IP (Integer Programmng)
78 JOURNAL OF COMPUTERS, VOL. 3, NO. 8, AUGUST 2008 has been appled [0]. Although, ths method can obtan the optmal soluton, t cannot satsfy the practcal response performance. The sweep method [] s a hgh-speed soluton adoptng comparatvely smple heurstcs. However, t s dffcult to solve a tough-constrant problem wth practcal accuracy. A mathematcal model such as a mnmum-cost-flow model wth a tme perod s also appled for constructng a large-scale transportaton network [2, 3, 4]. Ths method s sutable for determnng the transportaton capacty of each sub-network and for determnng the allocaton of the cost to each enterprse. However, t s mpossble to evaluate the accurate logstcs cost based on the number of used vehcles. Meta-heurstcs such as the tabu search or SA (Smulated Annealng) [5] combned wth a local search method such as cross-opt [6, 7] have also been proposed, and these methods can obtan hghly accurate solutons [8, 9]. However, t s dffcult to apply these methods to our problem because a vehcle pcks up and dstrbutes parts at the same tme. Moreover, some parts are drectly transported between factores, that s, not through depots. Thus, we cannot apply these local search methods to our problem. A method of applyng GA (Genetc Algorthm) that ntroduces route-exchange genetc operaton has also been proposed [20]. Although, t can obtan extremely accurate solutons, the response performance s nsuffcent for problems wth strct constrants. Wth the above-descrbed problems n mnd, we propose a method of applyng GA to solve the long-dstance transportaton problem. Ths soluton method satsfes both nteractve response performance and human-expert-level accuracy. The plannng of the long-dstance transportaton network and ts techncal problems are descrbed n secton 2. In secton 3, the concept of the method that meets the above requrements s proposed. In secton 4, the detal mplementaton of the method s descrbed. Secton 5 shows the expermental results for VRP benchmark problems. In secton 6, an experment on the long-dstance transportaton problem s explaned. Secton 7 presents our conclusons. II. LONG-DISTANCE TRANSPORTATION NETWORK PLANNING PROBLEM A. Cooperatve Logstcs Network Prevously, parts supplers ndvdually delvered ther own parts to factores. However, the cost of procurements logstcs ncreased because of the ncrease n hgh-mx, low-volume producton [2]. For ths reason, to realze total optmzaton of the procurements network, enterprses that belong to a supply chan set up a cooperatve logstcs network usng composed of depots (delvery center) as shown n Fg.. Ths cooperatve logstcs network conssts of several mutual sub-networks such as parts-collecton networks coverng parts supplers and depots and a long-dstance transportaton network among factores and depots. The key features of the long-dstance transportaton network are Parts collecton sub-networks Long-dstance transportaton sub-network Factory Factory Factory Fgure. Structure of cooperatve logstcs network. as follows. () Wde-area transportaton Factores and depots are located across a large natonwde area. Long-dstance transportaton s thus necessary for ths cooperatve logstcs. (2) Strct tme constrants Transportaton between depots and factores requres synchronzaton of the producton plans of each factory. Ths parts transportaton must therefore satsfy strct tme constrants. (3) Coexstence of pckup and delvery transportaton In ths network, parts are transported from depots to factores. Moreover, sem-fnshed products are transported from one factory to another. Vehcles thus pck up parts and delver parts smultaneously. (4) Mult depots There are several depots n the long-dstance transportaton network, and each depot has several vehcles. A vehcle starts from a depot and travels to several factores and other depots. Fnally, the vehcle returns to the orgnal depot. Along ths travelng route, some parts are drectly transported from a factory to other factores, that s, not through any depots. B. Techncal Problems Regardng Long-dstance Transportaton Network Plannng To mplement the schedulng method for the long-dstance transportaton network, the followng techncal problems must be addressed. () Human-expert-level optmalty for strct-constrant problems An optmal soluton for solvng the schedulng-phase problem s not always requred. However, to avod crucal error at estmaton of cost and operablty, at least, human expert-level optmalty s requred. (2) Response performance for nteractve operaton To realze a human expert s evaluaton from many aspects, response performance enablng nteractve operaton at least once or twce an hour s necessary. C. Model for Long-dstance Transportaton Network The schedulng problem s modeled, wth the followng
JOURNAL OF COMPUTERS, VOL. 3, NO. 8, AUGUST 2008 79 notatons, as follows. Q: maxmum loadng capacty of a vehcle. M: maxmum usable vehcles. {L } =..N : a set of loads. {w,.., w N, w N+,.., w 2N }: a set of pckup and delvery works. w 2- : pckup work of load L. w 2 : delvery work of load L. Q : amount of load that s pcked up or delvered at work w. When work w s a pckup work, Q s a postve value. On the contrary, Q s negatve, when work w s a delvery. Q satsfes (). Q = Q, Q > 0, Q < 0. () 2 2 2 2 {P } =..P : a set of transportaton ponts (e.g., factores and depots). And Pl(w ) means the transportaton pont where work w s done. [e, l ] : A tme constrant for work w. A vehcle to whch work w s assgned must arrve to delvery pont Pl(w ) no later than l. When the vehcle arrves before e, t must wat the start of the work (pckup or delvery) untl e. M : number of vehcles that the dstrbuton pont P holds. M means the total of all M as (2). P M = M. (2) = A vehcle route s a sequence of works as follows. W = w, w,.., w 2 n ( P l( w ) = Pl( w )) n. (3) A vehcle starts from a depot and travels several ponts (factores and depots) to pck up or delver parts. The vehcle fnally comes back to the orgnal depot. Thus, a travelng route of a vehcle s represented as a sequence of works (pckup or delvery). wk W k -: the set of works before n the sequence. W k +: the set of works after work w n the sequence. k Wrk(W ) : the set of works contaned vehcle route W. WORK: the set of all works. t(p,p j ): vehcle travelng tme from ponts P to P j. tm(w ): total travelng tme of vehcle route W. n tm( W ) = t( Pl( w ), Pl( W )) + t( Pl( w ), Pl( W )). (4) j j+ n j= ω={w, W 2,,W m }: a transportaton plan, where ω s a set of vehcle routes that satsfy constrants (5) and (6). m U Wrk( W ) = WORK. (5) = WrkW ( ) I WrkW ( j) = ( j ). (6) Moreover, a vehcle route W that s contaned n ω must satsfy the followng constrants. (a) Lmtaton on number of vehcles. The number of the vehcle that starts from the depot must not exceed the number of vehcles held by the depot. (b) Tme constrant Let W j be a vehcle route that work w s assgned to, and be the arrval tme of the vehcle to Pl(w ). The arrval tme must not exceed tme constrant l. a j a j l. (7) (c) Vehcle-total-travelng-tme constrant tm(w )<T. (8) (d) Pck-up-and-delvery-work-order constrant The correspondng pck-up work and delvery work must be assgned to the same vehcle by the followng order (9). w2j W w w 2 j 2j +. (9) (e) Load-capacty-constrant for a vehcle j 0 Q Q ( j n). (0) k k = α: fxed cost for a vehcle. c(p, P j ) : vehcle transportaton cost from ponts P to P j. The vehcle travelng cost of route W s represented as (). n tc( W) = α + c( Pl( w ), Pl( w )) + c( Pl( w ), Pl( w )). () j= j j+ n The total cost of transportaton plan ω s thus represented as (2). m tc( ω) = tc( W ). (2) = III. CONCEPT OF THE PROPOSED SOLUTION METHHOD FOR THE COOPERATIVE LOGISTICS NETWORK SCHEDULING A. Concept of Selfsh-constrant-satsfacton GA To guarantee both response performance and optmalty n solvng the transportaton-plannng problem as descrbed n secton 2, we propose a GA method, whch we call selfsh-constrant-satsfacton GA. Ths method bascally conssts of the followng two deas. The frst dea s a new concept of a selfsh gene, whch s dfferent from Dawkns' selfsh gene. The orgnal selfsh gene proposed by R. Dawkns [22] les n the followng dea. Namely, all genes of an ndvdual cooperate wth each other, amng at the prosperty of the ndvdual for ts selfshness. Strctly speakng, all genes consttutng a chromosome cooperate wth each other, amng at the prosperty or reproducton of the chromosome to satsfy the chromosome's selfshness. We further developed ths dea; that s, each gene on a chromosome or an ndvdual acts selfshly to optmze only tself or the relatonshp between tself and ts surroundngs locally, wthout consderng the totalty of the ndvdual or other genes of the same ndvdual. When the constrant s qute severe and hgher optmalty s requred, t s natural to magne that such a local selfsh acton occurs. The second dea s a concept of lmted allowance. Namely, to some degree, a dvne beng, namely, God, or a natural system admts reproducton or prosperty of a chromosome or an ndvdual, each consttuent of whch s a selfsh gene. However, n the end, through reflecton or punshment at some approprate generatons, only ndvduals that satsfy the objectves or constrants can survve over generatons.
80 JOURNAL OF COMPUTERS, VOL. 3, NO. 8, AUGUST 2008 We came up wth the above two deas n order to fnd an effcent GA method that can optmally solve strct-constrant problems. Concretely speakng, n the GA method for solvng strct-constrant transportaton schedulng problems, each gene satsfes only the tme constrants drectly related to tself, n other words, t does not consder satsfyng the tme constrants of other genes n the same chromosome. Further, each such chromosome, each consttuent of whch s a selfsh gene, s allowed to survve over generatons under the followng condtons. That s to say, the chance of globally satsfyng constrants s gven through changng the structure of each consttutng gene per executon of GA s operaton such as crossover and mutaton. Furthermore, an ndvdual (strctly speakng, a chromosome or a unt genes' group) s not klled untl some partcular generatons even f t ncludes a selfsh gene that cannot globally satsfy constrants. In ths way, by allowng the selfshness of each consttutng gene and not examnng the global constrants satsfacton except for some partcular generatons, GA s executon tme s shortened. Moreover, by allowng varous types of ndvduals to survve, more optmal solutons are found. Both hgh optmalty and speedy responsveness are thus obtaned. In the real world of lvng thngs, a group of unt genes or an ndvdual that perfectly adapts to the envronment does not suddenly emerge n ther evolvng stages. Prmtve and mperfect lvng thngs, whch surely volate some rules or constrants, evolve slowly to adapt to ther envronment. Lkewse, ths phenomenon exsts n the optmzaton of GAs. If we try to contnue satsfyng all constrants at each generaton, the populaton sze often shrnks or the system falls nto a local mnmum. Effcent optmzaton cannot be attaned n ths fashon. When constrants are very severe, n the worst case, the whole populaton can be destroyed. Ths s to say, f a populaton conssts of only ndvduals that satsfy constrants, a remarkable mprovement s not expected. However, there s a rsk of fallng nto a local mnmum. For such reasons, we also gve a lmted chance of survval, reflecton or reparaton to the ndvduals that volate constrants. B. Penalty for Tme-constrant Volaton and Reparaton When we permt the exstence of ndvduals that volate tme constrants, t s natural that all ndvduals wll eventually volate the tme constrants. To avod ths, a penalty s added to the ftness value for an ndvdual that volates the tme constrants. Concretely speakng, the value n whch the penalty coeffcent s multpled by the cost that corresponds to the delay tme s added to the cost estmaton. Moreover, an ndvdual that volates tme constrants s repared at constant generaton ntervals. To repar the tme-constrant volatons, pars of pckup and delvery works on a vehcle route that volate the constrant are removed sngly. After these removals are performed for all vehcle routes, the removed pckup and delvery works are renserted nto a vehcle route that globally satsfy tme constrants. If there s no vehcle that can satsfy the tme constrants, a new vehcle s allocated to the transportaton. C. Dynamc Penalty Control The above mentoned penalty s regarded as lattude for ndvduals that volate tme constrants. The large lattude promotes the convergence of the objectve functon. On the other hand, the small lattude prompts a search n the constrant satsfacton solutons. However, t s dffcult to generate new ndvduals that satsfy all constrants for hard constrant problems. To avod ths problem, we apply a dynamc penalty control method [23]. Ths method promotes the optmzaton of the objectve functon by usng relatvely large lattude at start tme. Then, the lattude s decreased gradually n order to reduce the ndvduals that volate tme constrants n the populaton. In more detal, the penalty s ncreased exponentally from ntal value at each generaton. At the reparaton generaton, the penalty s turned down to the ntal value. D. Constrant Pre-checkng Method To obtan hghly optmal solutons by avodng the convergence nto the local mnmum, the GA must have enough populaton and generatons. However, long chromosomes contanng many genes are needed to represent our plannng problem. Therefore, both the mutaton operaton and the crossover operaton take a long calculaton tme. Moreover, the calculaton slows down because varous constrants on our proposed GA are checked. To avod such problems caused by varous constrants, we propose a constrant pre-checkng method. Ths method ntroduces nto our GA the concept of constrant propagaton utlzed as an effcent solvng method for constrant-satsfacton problems [24, 25]. Namely, ths method statcally analyses the constrants on the problem, before startng the GA. Ths reduces the overhead for dynamcally searchng nodes that cannot be vsted by one vehcle. Ths mproves the calculaton effcency of routes reconstructon durng the crossover and the mutaton of GA. That s, constrants brought by, for example, dstrbuton tme range, load capacty, and node s geographcal poston are statcally analyzed before startng the GA and set n a table of exclusve nodes. Thus, durng the executon of GA, the system does not need to dynamcally repeat the check of these constrants. In partcular, because dstrbuton ponts (factores and depots) are spread over a naton-wde area, the geographcal condtons effectvely reduce the combnaton of ponts that can be vsted by one vehcle. Snce there are many constrants on our long-dstance transportaton plannng problem, t s expected that ths method reduces the calculaton tme of the GA and realzes near-real-tme response performance. In the operaton of our cooperatve logstcs network, a vehcle does not go to a far pont (depot or factory) durng runnng neghborng ponts. Ths operatonal rule s represented by the followng constrants (3), (4), and (5). The x, y, and z represent transportaton ponts. When these three condton are satsfed, the pckup or delvery work at
JOURNAL OF COMPUTERS, VOL. 3, NO. 8, AUGUST 2008 8 y cannot be nserted nto a vehcle route between x and z. l 0 <t(x,z)<l. (3) t(x,y)+t(y,z)-t(x,z)>l 2. (4) t(x,y)+t(y,z)-t(x,z) >βt(x,z). (5) β,l 0, l, and l 2 are fxed values. IV. IMPLEMENTATION OF THE PROPOSED METHOD A. Structure of Chromosome The soluton to the long-dstance transportaton problem ncludes clusterng of loads, whch ndcate ther correspondence to vehcles, and ther travel routes n each cluster. To express both clusterng and routng nformaton, the chromosome structure shown n Fg. 2 (a) s appled. The chromosome s a sequence of work IDs of pckups and delveres. Table shows a detal of the load nformaton. The load nformaton conssts of ts orgn, destnaton, pckup work ID, and delvery work ID. The load s transported from P to P5, and ts pckup d s, and delvery work d s 2. In a smlar manner, the load 2 s transported from P3 to P4. In ths example, the frst vehcle of the chromosome n Fg. 2 (a) travels round P3, P4, P6, and P8 as shown n Fg. 2 (b). B. Schedulng Algorthm usng GA The selfsh-constrant-satsfacton GA constructs solutons dsregardng global constrants. Ths type GA generates ndvduals that volate tme constrants n the populaton. Consequently, to solve ths problem, these ndvduals have to be repared at constant generaton ntervals. Moreover, dynamc penalty control s also appled for effcent optmzaton. Thus, n addton to the normal genetc operatons such as selecton, crossover, and mutaton, reparaton for ndvduals that volate tme constrants and recalculaton of the penalty are performed as follows. Step. Generate ntal populaton. Step 2. Calculate penaltes. Step 3. Calculate ftness value for each ndvdual. Step 4. Perform selecton. Step 5. Perform crossover. Step 6. Perform mutaton. Step 7. Repar ndvduals that volate tme constrants. Step 8. Check termnates condton. Step 9. Go to step 2. TABLE. SAMPLE PICKUP AND DELIVERY WORK OF LOAD ID Orgn Destnaton Pckup Delvery work ID work ID P P5 2 2 P3 P4 3 4 3 P3 P6 5 6 4 P5 P9 7 8 5 P4 P8 9 0 6 P P2 2 3 5 4 9 6 0 2 Load: 2 P3 P4 (a) Vehcle Vehcle 2 Load: 3 P3 P6 Sample representaton of chromosome Vehcle : P3 P4 P6 P8 Vehcle 2: P P2 P5 (b) Load: 5 P4 P8 Load: P P5 Sample vehcle round routes Fgure 2. Sample representaton of chromosome and vehcle routes. C. Ftness Value and Selecton The ftness value of the ndvdual becomes larger, when the total cost becomes smaller. To be more precse, let U s an enough large fxed value compared to the total cost, and the ftness value s defned as (6). To create a next generaton, ndvduals are selected at a rate proportonal to ther ftness value. In addton, the elte ndvdual who has the hghest ftness value s always copy to the next generaton. U-tc(ω). (6) D. Crossover and Mutaton To realze the above concept of selfsh-constrantsatsfacton GA, an nserton method called selfsh NI (Nearest Inserton) s appled to put nodes nto a vehcle route. Wth ths method, a node s nserted nto a tour, usng NI method, f the node satsfes ts own constrants only. That s, the nfluence on the other nodes s not consdered. The crossover constructs a chld from two parents accordng to the followng process. () Determne the crossover pont n one parent chromosome. (2) Obtan vehcle routes represented by a group of genes located before the crossover pont n the chromosome. (3) Change the order of remanng nodes that are not contaned n any vehcle routes obtaned n (2) accordng to the order of nodes (genes) n the other parent s chromosome. (4) Insert the remanng nodes nto the route obtaned n (2) by usng the selfsh NI method n the order reordered n (3). (5) If no vehcle satsfes the tme and load-capacty constrants, a new vehcle s assgned to the transportaton of the load. In ths crossover process, when a pckup node s deleted from a vehcle route, the correspondng delvery node s also removed. Namely, a par consstng of a pckup work and a delvery work s deleted and nserted nto the same 2 Load: 6 P P2
82 JOURNAL OF COMPUTERS, VOL. 3, NO. 8, AUGUST 2008 vehcle route smultaneously. The mutaton randomly selects a mutaton node and deletes t together wth ts neghborng nodes. The deleted delvery ponts are renserted nto the transportaton plan usng the selfsh NI method. Ths delete and rensert process s done for each load. The pckup/delvery work par s always assgned to the same vehcle. E. Reparaton The selfsh-constrant-satsfacton method permts to survve ndvduals who volate tme constrants. Thus, these ndvduals should be repared at constant generaton ntervals by the followng two steps. () Elmnaton of loads that volate tme constrants If a pckup or delvery tme constrant s volated, both the pckup and delvery works are removed from a vehcle route. Accordngly, the chromosome shortens by ths elmnaton. (2) Renserton of elmnated load The removed pckup and delvery works are renserted nto a vehcle route whch satsfes constrants globally. When there s no vehcle whch satsfes all constrants, a new vehcle s assgned to the schedule and the elmnated pckup and delvery works are nserted nto the newly assgned vehcle. Namely, the pckup and delvery works are added to the end of the chromosome n Fg. 2 (a). F. Proposed Soluton Method We propose the followng two method usng the above mentoned elemental soluton methods. () Selfsh-constrant-satsfacton GA wth fx penalty Ths soluton method constructs a vehcle route usng the selfsh NI method and permts the exstence of ndvduals that volate constrants n the populaton. The penalty value s fxed for all generatons n ths method. (2) Selfsh-constrant-satsfacton GA wth dynamc penalty control Ths soluton method also permts the exstence of ndvduals that volates tme constrants. Moreover, the penalty for ndvduals that volates tme constrants s dynamcally modfed. In the next secton, experments to evaluate the effect of the above two selfsh-constrant-satsfacton GAs are performed. These tests compare them wth Non-selfsh GA. Non-selfsh GA apples conventonal non-selfsh NI method to crossover and mutaton operaton. Ths NI method nserts a node nto the poston that satsfes not only ts constrants but also the constrants of every other node n the same ndvdual. In case of such postons of nodes are not searched out, crossover or mutaton operaton assgns a new vehcle. V. EXPERIMENTS USING THE BENCHMARK PROBLEM To evaluate the proposed selfsh-constrant-satsfacton GA, we determned the penalty value for tme-constrant volatons, and the generaton gap nterval for the reparaton usng VRP benchmark problems [26]. We then compared the accuracy of the selfsh-constrant-satsfacton GA wth that of the global-constrant-satsfacton GA, whch satsfes all constrants all the tme, by these benchmark problems. In the next secton, to evaluate the proposed method, we appled t to the long-dstance-transportaton-network plannng problem. GA parameters of populaton sze, mutaton rate, and crossover rate were determned based on an exploratve experment. In ths experment, we executed GAs n 30 mnutes and compared solutons. As a result, we determned the populaton sze, mutaton rate, and crossover rate as 00, 5%, and 0% respectvely. A. Experment Parameters () Penalty coeffcent for tme-constrant volaton Our target long-dstance transportaton network contans about one hundred dstrbuton ponts (factores and depots). Thus, we selected a VRP benchmark problem wth 00 dstrbuton ponts to determne the parameter for our proposed GA. The accuracy of the soluton accordng to the change n the coeffcent s shown n Fg. 3. When there s no penalty, the error rate s over 0%. When the coeffcent s larger than 3, the error rate stays under 5%. In partcular, the error rate becomes less than 3% when a value from 5 to 0 s specfed for the coeffcent. Accordng to ths result, we determned the coeffcent as 5. (2) Generaton gap for gene reparaton Fg. 4 shows the change n the accuracy accordng to the change n the generaton gap nterval of the tme-constrant reparaton. The error rate s over 3% when the gap s less than 3. The accuracy becomes a mnmum when the gap s between 5 and 0. We thus determned the gap as 0 generatons. Error (%) 4 2 0 Fgure 3. 8 6 4 2 0 0 0 20 30 40 50 Penalty coeffcent Correlaton between penalty coeffcent and accuracy.
JOURNAL OF COMPUTERS, VOL. 3, NO. 8, AUGUST 2008 83 Error (%) 0 8 6 4 2 0 0 0 20 30 40 50 Generaton nterval for reparaton TABLE 2. ACCURACY OF EACH METHOD FOR PROBLEM Method Generaton Accuracy Non-selfsh GA,000 4.8% Selfsh-constrant-satsfacton GA wth fx penalty,000 2.5% Selfsh-constrant-satsfacton GA wth dynamc penalty control,000 2.5% TABLE 3. ACCURACY OF EACH METHOD FOR PROBLEM2 Method Generaton Accuracy Non-selfsh GA,000 6.7% Selfsh-constrant-satsfacton GA wth fx penalty,000 2.7% Selfsh-constrant-satsfacton GA wth dynamc penalty control,000 2.7% TABLE 4. ACCURACY OF EACH METHOD FOR PROBLEM3 Fgure 4. Correlaton between generaton gap and accuracy. B. Evaluaton of Accuracy To evaluate the accuracy of the soluton, we appled the proposed method to the VRP benchmark problems. Because of the tght connectons between the producton schedules of each factory, the long-dstance transportaton schedulng s assgned strct tme constrants. We prepared three problems: Problem, Problem 2, and Problem 3. Problem s the bench mark problem (R0) whch has the hardest tme constrants n bench mark problems. Problem 2 and Problem 3 have the same dstrbuton ponts and load nformaton as Problem. However, we reduced the tme wndows of Problem 2 to half of Problem. Moreover, Problem 3 has strcter tme-constrants than Problem 2. The tme wndows of Problem 3 are reduced to half of Problem 2. Thus, the tme wndows of Problem 3 are one fourth of Problem. These three problems have the same optmal soluton. In ths experment, we solve the problem usng the followng three methods. () Non-selfsh GA (2) Selfsh-constrant-satsfacton GA wth fx penalty (3) Selfsh-constrant-satsfacton GA wth dynamc penalty control Table 2 lsts the accuracy of above three methods for the Problem n calculaton of 000 generatons (when the soluton s converged). Though the accuracy of the non-selfsh GA exceeded 4.5%, the selfsh-constrantsatsfacton GAs can obtan the soluton wth under 3.0% error. Table 3 lsts the accuracy for the Problem 2. The both selfsh-constrant-satsfacton GAs obtans solutons wth 2.7% error. Moreover, table 4 shows the results for the Problem 3. The selfsh GAs can obtan the better soluton than the non-selfsh GA as the result of Problem 2. In partcular, the thrd method can obtan the soluton wth under 4.0% error. Method Generaton Accuracy Non-selfsh GA,000 7.3% Selfsh-constrant-satsfacton GA wth fx penalty,000 4.2% Selfsh-constrant-satsfacton GA wth dynamc penalty control,000 3.8% C. Evaluaton of Response Performance To evaluate the constrant pre-checkng method that calculates the combnaton of ponts that the same vehcle cannot go around, we examned two methods: one uses constrant pre-checkng and the other does not use pre-checkng. The pre-checkng method bumps up the generatons that could be calculated wthn fve mnutes from 820 to 045 as shown n table 5. Moreover, t also mproves the accuracy from 2.9% to 2.6% error by the ncrease of the calculated generatons. VI. APPLICATION TO THE LONG-DISTANCE TRANSPORTATION PLANNING PROBLEM To evaluate the proposed method usng the same condton as the long-dstance transportaton problem, we prepared a transportaton problem wth mxed transportaton (.e., pckup and delvery) and many depots. Ths expermental network conssts of 70 factores and 30 depots. Moreover, 500 loads were prepared for the network. We set the orgn and destnaton, volume, and tme constrants for each load. The constant values, l 0, l, l 2, and β n (3), (4), and (5) were set as 0km, 00km, 700km, and 7 respectvely. Fg. 5 shows the convergence of the solutons. The proposed methods can calculate 300 generatons wthn 30 mnutes. Table 6 shows the solutons of 300 generatons GA. In ths experment the selfsh-constrant satsfacton GA wth fxed penalty can obtan about 0% less cost soluton than the non-selfsh GA. In addton, the cost s mproved 0.5% due to the dynamc penalty control method.
84 JOURNAL OF COMPUTERS, VOL. 3, NO. 8, AUGUST 2008 Cost 50000 45000 40000 35000 30000 25000 Non-selfsh Fxed penalty Dynamc penalty 2) The second dea s the dynamc penalty control method. Ths method promotes the optmzaton of the objectve functon by ncreasng the penalty for the ndvduals that volate tme constrants accordng to the reparaton cycle. 3) The thrd dea s constrant pre-checkng method. In our problems, the dstrbuton ponts dspersed to a large natonwde area. Thus, ths method statcally analyses the constrants on the problem before startng the GA to reduce the overhead for dynamcally searchng nodes that cannot be vsted by one vehcle. Our expermental results revealed that our proposed method enables nteractve operatons once or twce an hour for problems on a practcal scale. Moreover, ths method can obtan solutons wth human expert-level accuracy for hard constrant problems. 20000 5000 0000 0 50 00 50 200 250 300 Fgure 5. Generaton Convergence of solutons. TABLE 5. EFFECT OF CONSTRAINT PRE-CEHCKING METHOD Method Generaton Accuracy Non pre-checkng 820 2.9% Pre-checkng,045 2.6% TABLE 6. ACCURACY OF SOLVINE LONG-DISTANCE TRANSPORTATION PROBLEM Method Cost Non-selfsh GA 3,52 Selfsh-constrant-satsfacton GA wth fx penalty 2,32 Selfsh-constrant-satsfacton GA wth dynamc penalty control 2,065 VII. CONCLUSION We proposed a plannng method for a long-dstance transportaton network that obtans both nteractve response performance and human expert-level accuracy. Ths soluton method s based on the followng three deas. ) Frst dea s selfsh-constrant-satsfacton GA. Each gene, whch consttutes an ndvdual, does not care about constrants of all other genes wthn the same ndvdual. Moreover, an ndvdual s not klled untl some partcular generatons even f t ncludes a selfsh gene that cannot globally satsfy constrants. Ths method enables to avod fallng nto a local mnm for strct-constrant problems. REFERENCES [] T. P. Stank and P. J. Daugherty: "The mpact of operatng envronment on the formaton of cooperatve logstcs relatonshps", n Transportaton Research Part E: Logstcs and Transportaton Revew, Vol. 33, No., pp. 53-65, March 997. [2] M. Kubo, Logstcs engneerng, Tokyo: Asakura-Syoten, 200. [3] M. Wasner and G. Zapfel: An ntegrated mult-depot hub-locaton vehcle routng model for network plannng of parcel servce, n Int l Journal of Producton Economcs, Vol. 90, No. 3, pp. 403-49, Aug. 2004. [4] T. Onoyama, S. Kubota, K. Oyanag, and S. Tsuruta, A method for solvng nested combnatoral optmzaton problems a case of optmzng a large-scale dstrbuton network, n Proc. IEEE SMC 2000, pp. 2467-2472, Oct. 2000. [5] T. Onoyama, T. Maekawa, S. Kubota, and N. Komoda, Hgh-speed plannng method for cooperatve logstcs networks usng mxed nteger programmng model and dummy load, n Trans. Electrcal Engneerng n Japan, Vol. 64, No. 2, pp. 64-70, Feb. 2008 [n Trans. IEEJ, Vol. 27-C, No., pp.3-36, Jan. 2007]. [6] H. J. Sebastan: Strategc plannng of dstrbuton networks for letter and parcel mal usng optmzaton models, n Abstract of the 9th Int l Workshop on Complex Systems Modelng jontly wth the 6th Int l Symposum on Knowledge and Systems Scences (CSM/KSS 05), pp. 20-2, Aug. 2005. [7] J. Bramel and D. Smch-Lev: The Logc of Logstcs, New York: Sprnger-Verlag (997). [8] M. Gendean, G. Laporte, and J.Y. Potvn, Vehcle routng modern heurstcs, n E. Aarts and J. K. Lenstra (Eds.), Local Search n Combnatoral Optmzaton, Prnceton: Prnceton Unversty Press, pp. 3-336, 997. [9] T. Onoyama, T. Maekawa, S. Kubota, S. Tsuruta, and N. Komoda, Vehcle routng problem solvng method for a cooperatve logstcs network by usng mult-stage GA, n Trans. IEEJ, Vol. 27-C, No. 9, pp. 460-467, Sep. 2007. [0] P. Toth and D. Vgo: Exact soluton of the vehcle routng problem, n T.G Cranc and G. Laporte (Eds.), Fleet Management and Logstcs, Kluwer Academc Publshers, pp. -3, 998. [] T. Masu, S. Yurmoto, and N. Katayama, OR n Logstcs, Tokyo: Maksyoten, 998.
JOURNAL OF COMPUTERS, VOL. 3, NO. 8, AUGUST 2008 85 [2] H. Mohr, M. Kubo, and M. Mor, Dstrbuton plannng problem on manlne transportaton, n Abstract of Japan OR Socety 995, pp. 28-29, Sep. 995. [3] S. Ataka and M. Gen, Soluton method of mult-product two-stage logstcs problem wth constrants of delvery course, n Trans. IEEJ, Vol. 28, No. 3, pp. 456-46, March 2008. [4] H. Mohr, T. Watabnabe, M. Mor, and M. Kubo: Cost allocaton on vehcle routng problem, n Journal of the Operatons Research Socety of Japan, Vol. 40, No. 4, pp. 45-465, Jan. 997. [5] S. M. Sat and H. Youssef: Iteratve Computer Algorthms wth Applcatons n Engneerng: Solvng Combnatoral Optmzaton Problems, Los Alamtos: IEEE Computer Socety, 2000. [6] N. Park, H. Okano, and H. Ima, A path-exchange-type local search algorthm for vehcle routng and ts effcent search strategy, n Journal of the Operatons Research Socety of Japan, Vol. 43, No., pp. 97-208, Jan. 2000. [7] E. Tallard, P. Badeau, F. Guertn, and J.-Y. Potvn: "A tabu search heurstc for the vehcle routng problem wth soft tme wndows", n Transportaton Scence, Vol. 3, No. 2, pp. 70-86, Feb. 997. [8] M. Gendreau, H. Alan, and G. Laporte: A tabu search heurstcs for the vehcle routng problem, n Management Scence, Vol. 40, No. 0, pp. 276-290, Oct. 994. [9] J. H. Hooker and N. R. Natraj: Solvng a general routng and schedulng problem by chan decomposton and tabu search, n Transportaton Scence, Vol. 29, No., pp. 30-44, Jan. 995. [20] G. A. P. Kndervater and M. W. P. Savelsbergh: Vehcle routng: handlng edge exchanges, n E. Aarts and J. K Lenstra (Eds.), Local Search n Combnatoral Optmzaton, Prnceton Unversty Press, pp.337-360, 997. [2] H. Kuse: Logstcs and socal problems, n Journal of the Socety of Instrument and Control Engneers, Vol. 37, No. 3, pp. 66-69, March 998. [22] R. Dawkns, The Selfsh Gene, London: Oxford Unversty press, 989. [23] Y. Sakura, T. Onoyama, S. Kubota, Y. Nakamura, and S. Tsuruta, Locally selfsh-gene tolerant dynamc control GA to solve constrant TSPs, n Trans. IPSJ, Vol. 48, No. SIG-9, pp. 27-38, Dec. 2007. [24] V. Kumar: Algorthms for constrants satsfacton problems: A survey, n AI Magazne, Vol. 3, No., pp. 32-44, Jan. 992. [25] P. Shaw: Usng Constrant Programmng and Local Search Methods to Solve Vehcle Routng Problems, Berln: Sprnger-Verlag, 998. [26] M. M. Solomon: VRPTW benchmark problems. http://web.cba.neu.edu/~msolomon/problems.htm Dr. Onoyama s a member of the IEEE and others. Takuya Maekawa was born n 97. He receved the Bachelor of Scence from the Yamagata Unversty n Japan n 993. He joned Htach Software Engneerng Co., Ltd n 993 and he has been developed schedulng system. Hs man research nterests are VRP solvng method for logstcs plannng. Mr. Maekawa s a member of the IPSJ. Yoshtaka Sakura was born n 977. He receved Bachelor of Informaton Scence from the Unversty of Electro- Communcatons n 2000. He receved the PhD n Engneerng from the Unversty of Electro-Communcatons n 2005. He teaches at the Tokyo Denk Unversty from 2005. Hs man research nterest s theory and applcaton of evolutonal and learnng algorthms. Dr. Sakura s a member of the IEEE and IPSJ. Setsuo Tsuruta was born n 947. He receved the Bachelor from the Waseda Unversty n 97 and he completed the Master of Engneerng at the Nagoya Unversty. He receved the PhD n Engneerng from the Nagoya Unversty. He joned Htach Ltd n 973. In 2003, he joned the nformaton envronment faculty of Tokyo Denk Unversty as a professor. Hs research focuses on the schedulng system and ntellgent system. Prof. Tsuruta s member of IEEE and others. Norhsa Komoda was born n 950. He completed the Master of Engneerng at the Osaka Unversty n 974. He receved the PhD n Engneerng from the Osaka Unversty n 98. He joned Htach Ltd n 974. In 99 he joned the engneerng faculty of Osaka Unversty as assocate professor. Snce 992 he s professor of Graduate School of Informaton Scence and Technology of the Osaka Unversty. He s manly nterested n the nformaton system n the area of manufacturng and dstrbuton ndustry and knowledge based nformaton processng. Prof. Komoda s a member of IEEE and others. He receved 998 IEEJ Paper Award. Takash Onoyama was born n 959. He receved the Bachelor of Scence from the Osaka Unversty n 98. He receved the PhD n Informaton Scence from the Osaka Unversty n 2007. He joned Htach Software Engneerng Co., Ltd n Tokyo, Japan n 98. In 2003, he became Department Manager of Research & Development Department. The focus of hs research s concentrated wthn the topcs of plannng method for logstcs system.