Approaches Method to Solve Ships Routing Problem with an Application to the Indonesian National Shipping Company

Transcription

1 Approaches Method to Solve Shps Routng Problem wth an Applcaton to the Indonesan Natonal Shppng Company ISMAIL YUSUF 1,2), ACHMAD YANI 2,3), and MOHD. SAFIYAN BABA 1) Department of Artfcal Intellgence, Faculty of Computer Scence and Informaton technology. Unvety of Malaya. Malaysa 1) LaMntang Educaton and Tranng (LET) Batam. Perum Bandaramas Blok D5 No.1. Batam KEPRI. Indonesa 2) Department of Shppng Engneerng, Faculty of Engneerng, Hasanuddn Unvety Makassar, South Sulawes Indonesa 3) Abstract: - In archpelagc countres wth long shorelnes or wth many wde rve, water transportaton may play a sgnfcant role also n domestc trades, e.g. Indonesa. Indonesa s an archpelago that ncludes roughly 17,508 slands where the total area s 741,052 square mles and wth an area of ocean s 35,908 square mles. The Indonesan archpelago stretches 3181 mles from east to west (Longtude: 97ºE E) and 1094 mles from north to south (Lattude: 6 N - 11ºS). Indonesa needs a system of sea-based for nter-sland transportaton that can assst n overcomng solaton arsng from geographc dfferences. Sea-based transportaton systems depend on effectveness of the route and shps. Ths research proposes a method called CAB-Method and help to fnd a set of routes gvng mnmal total travel cost satsfyng some constrans are spesfcaton of the shps, and sea-depth n the ports. From the result presented shows that route made by CAB-Method wll ncrease the performance mprovement of PT. PELNI (PT. PELayaran Nasonal Indonesa = the Indonesan Natonal Shppng Company) by save the total cost about $US 75, for serve all port n Indonesa and also wth more number port vsted, compared exstng route. Increasng number port vsted wll ncrease the number of passenge who could be served. Key-Words: - CAB-Method, PELNI, Vehcle Routng Problem, Shp, Transportaton 1 Introducton Transportaton s a basc of develop the people and economc, and ncreasng the growth of naton ndustry. Geographc locaton of a regon wll affect the effcent transportaton system. In archpelagc countres wth long shorelnes or many wde rve, water transportaton play a sgnfcant role and also n domestc trades, e.g. Phlppnes, Indonesa, and Japan. And for the wder stuaton, water transportaton s the major condut of nternatonal trade. It s causes the share of ts weght borne by sea. The need transport sea wll cause the shp as well as hs supporter (port, route and management) to have to be ncreased [1, 9, 10]. Shps come n a varety of types who each made for dfferent uses. Shps operate between ports. Ports are used for loadng and unloadng cargo/passenger as well as for loadng fuel, fresh water and supples, and dschargng waste. Ports mpose physcal lmtatons on the dmensons of the shps (shp draft, length and wdth), and usually charge fees for ther servces. In order to be able to make strategc decsons one usually needs some tactcal or even operatonal nformaton. Models used for fleet sze and network desgn decsons often requre evaluaton of shp routng strateges. Such routng models usually fall nto one of two categores, arc flow models or path flow models [10]: 1. In arc flow models a bnary varable s used to represent whether a specfc vessel v travels drectly from port (or customer) to port (or customer) j. The model constructs the routes that wll be used by the vessels and the model have to keep track of both travel tme and load on each vessel. 2. In path flow models the routes are predefned, one way or another and a bnary varable represents whether vessel v performs route r. A route s usually a full schedule for the vessel that specfes expected arrval tmes and load on the vessel along the route. Such a model can focus on the set of ports or custome to serve, and only feasble routes are consdered. ISBN:

2 Shppng routes may be classfed accordng to ther geographcal characterstc of the ports (e.g. sea-depth); and the correspondng type and sze of vessel used (e.g. shp draft and speed). Desgn route for path flow model seen as VRP. The VRP arses naturally as a central problem n the felds of transportaton, dstrbuton, and logstcs [6]. The VRP may actually be consdered a broad class of routng problems, composed of many specfc varants of the basc defnton. Some of these VRP varants and ther unque constrants are presented below [3, 5, 12]: 1. Capactated Vehcle Routng Problem (CVRP), VRP wth the addtonal constrant that all vehcles wthn the fleet have a unform carryng capacty of a sngle commodty. The commodty demand along any route assgned to a vehcle must not exceed the capacty of the vehcle. a. Heterogeneous CVRP (or Mxed Fleet CVRP), the fleet s composed of dfferent vehcle types. b. Homogeneous CVRP (or Unform Fleet CVRP), each vehcle n the fleet has the same capacty. 2. Ste Dependent Capactated VRP (SDCVRP), s a varant of the Heterogeneous Capactated VRP where there exsts a dependency between the type of vehcle and the customer, meanng that not every type of vehcle can serve every type of customer because of ste-dependent restrctons. 3. Multple Depots VRP (MDVRP). Note that these varants do not necessarly exclude each other, combnatons of two or more of these varants can be made to form more complex varants of the VRP. 2 Problem Descrpton Ths study s a combnaton of SD-CVRP and multple depots by addng constran, the fuel port. Whch varants of the VRP wth multple depots are referenced to as MD-VRP. Snce the fleet s composed of dfferent vehcle types (Heterogeneous CVRP) then there exsts a dependency between the type of vehcle and the port, meanng that not every type of vehcle can serve every type of port because of ste-dependent restrctons (Ste Dependent CVRP). The problem soluton conssts n determnng a feasble set of routes R ps served by shps s, wth constran: 1. The objectve functon s expected to mnmze the total voyage cost 2. All the ports should be served 3. Each port s served by mnmum a shp. 4. Each shp s assgned exactly once by exactly one route. Let, P d j S s a port set, and each port (p P) wth a dfferent geographcal characterstc s sea-depth, h p n meter. P port set has some F fuel port, F. s a departure port and j s a arrve port. s denote the dstance from port to port j n nautcal mles, and travel from port to port j does matter, so d j d j for all port and j. s a shp set, and each shp (s S) has a dfferent shp drafts (δ s n meter) and speeds (V s n knot). R ps s a set of feasble route for port p, whch port p s feasble served by shp s wth hard constrans: 1. The path between all ports n a feasble route for shps s (called a total voyage) of any route does not longer than D, D s the total travel tme allowed. 2. Shp s wth shp draft δ wll serve route r, wthout exceed the sea-depth n any port n R ps. 3. The set of feasble route R ps has mnmum one fuel port. 3 Model Formulaton The classcal VRP can be defned as follow [8], let G = (V, A) be a graph where V = {1, n} s a set of vertces representng ctes wth the depot located at vertex 1, and A = {(v, v j ) v, v j V, < j} s the set of arcs. Wth every arc (, j) s assocated a non-negatve dstance matrx C = (c j ). In some contexts, c j can be nterpreted as a travel tme or a travel cost. In order to present the mathematcal formulaton of the models, needs to follow notatons: S set of shps to be scheduled, ndexed by s P set of ports, ndexed by, j, p R set of route, ndexed by r R ps set of feasble route cluste for shp s to serve port p wthout exceed the constrans, ndexed by r ps R s set of route cluste for shp s wth leavng port, ndexed by r s NS Number of shps to be scheduled. NP Number of ports. NR ps Number of feasble routes for shp s to serve port p wthout exceed the constrans. F fuel port, F. Paramete: Cost(R ps ) s the cost of assgnng a shp s to route cluster R ps. Y rs s a bnary varable that equal to 1 f route r served by shp s and leavng a port ; and 0 otherwse. ISBN:

3 Varable: s a bnary varable that equal to 1 f route r served by shp s and t s selected n the soluton; and 0 otherwse. mnmze Cost(R ps ) (1) ss rr rr kk Y rs ss 1, (2) 1, s S (3),1, s S, r Rps 0 (4) Constrant (1) the objectve functon s mnmzes the transportaton costs; Constrant (2) ensure that all ports are servced; Constrant (3) ensure that route r served by shp k and t s selected n the soluton; and Constrant (4) mposes bnary requrements on the varables. In order to present mathematcal formulaton of the constran model that needs the followng notaton: α Penalty assocated wth constran volaton respect to the sea-depth and shp-draft. β Penalty assocated wth constran volaton respect to the route wthout any fuel port Snce the shp s composed of dfferent shp types then there exsts a dependency between the type of shp and the port, meanng that not every type of vehcle can serve every type of port. Ths study ntroduces a varable α that s 1 f the shp draft < sea-depth, and 0 otherwse. < h (5) ss s p 1, (6) Restrctons (5) and (6) ensure that the shp-draft of the shps should not same or exceed the sea-depth of the ports. The other constran s route served by shps k should possess a fuel-port. Ths study ntroduces a varable β that s 0 f the route r has not a fuel port and 1 otherwse. F 1, F (7) f P rr 1, (8) {0, 1}, (9) {0, 1}, (10) {0, 1}, (11) Restrctons (7) and (8) ensure that the route r possess a fuel port. Whch (9), (10), and (11) are mposes bnary requrements on the varables. 4 CAB-Method The approxmate algorthm used to solve VRP. The approxmate algorthm manly ncludes two knds of algorthms, constructve heurstc algorthm and metaheurstc algorthm. Some research efforts were orented towards the development and analyss of approxmate heurstc technques capable of solvng real VRP problems. Bowerman et al. [15] classfed the heurstc approaches to the VRP nto fve classes: 1. Cluster-ft and route-second (CFRS), 2. Route-ft and cluster-second (RFCS), 3. Savngs and nserton, 4. Improvement and exchange, 5. Smpler mathematcal programmng representatons through relaxng some constrants. Ths study proposes 3-phases algorthm, CAB- Method: 1. Clusterng the ports based on total of the travel tme allowed 2. Assgn vehcles to the routes n conformance wth the restrctons on shps and ports. 3. Buld heurstc/meta-heurstc for optmal soluton. 4.1 Cluster Ports Anderberg [13] stated that a clusterng s a classfcaton technque to group a set of objects nto cluste such that the objects n the same cluster are smlar n some sense and those n dfferent cluste are dssmlar n the same sense. Cluster-Algorthm n ths phase used for solvng cluster problem based on soft constran, as example: load factor, travel dstance and other. Clusterng s performed to ensure the approprate port nto groups based on soft constrants. Phase 1: Let, P s a set of port {p1, p2, NP}, S s a set of shps {s1, s2, NS}, and R ps {r 11, r 21, NR ps } s a set of feasble route for shp s to serve port p wthout exceed the constrans. Phase 2: Startng wth the port, p n and shp, s n {n=1, 2,, n} For port cluster {Cluster(p,s) : p ϵ P, s ϵ S } Phase 3: Sort the port and keep nsertng the nearest unvsted port. Assgn port to new route wthout exceed the restrant, D. Phase 4: Calculate total D of the ports, whch (D = D n-1 + D n ): If the any ports of the route s not exceed the restran D, keep the port p n=1 that avalable for shp s n nto route cluster R ps and then the cycle ndex n=n+1 for the next port (p n+1 ), return to phase 3. ISBN:

4 Phase 5: Otherwse, leave the port p n=1 and then the cycle ndex n=n+1 for the next shp (s n+1 ), go to phase 2 for next route. Fnsh the algorthm untl all ports served by all shps, and ensure the approprate ports nto cluster groups R ps. The example s show n Fgure 1. Fgure 1. Cluste of the ports 4.3 Buld Heurstc or Meta-heurstc Buld heurstc or meta-heurstc based on two phases above (clusterng the ports and assgn vehcle) for the best soluton. Types of heurstc methods many exst [11, 12, 16] and there are some AI-orented methods [3, 5], whch can be roughly categorzed nto: 1. Constructon algorthms Start wth an empty route and extend t gradually, keepng the total cost as low as possble. 2. Tour Splttng Algorthms Start wth one bg tour and splt t nto capactyfeasble sub-tou. 3. Savngs based algorthms Start wth a lot of small routes and combne them as long as ths mproves the soluton ( combne and conquer ). Ths research used heurstc method to determne the optmal soluton of the problem and should ensure the restrants are not volated, that s: The combnaton of the selected routes (booths of cluste) s to have a lowest total cost wth all the ports should be served, and the combnatons of the routes wll served by all shps. 4.2 Assgn Vehcles The 2nd phase s assgn vehcle. Ths phase used for solvng cluster problem based on some constrants. It s performed to ensure the approprate of vehcles to serve avalable ports. And the set of feasble route R pk has mnmum one fuel port. Phase 1: Let, R ps {r 11, r 12, NR ps } s a set of feasble route for shp s to serve port p wthout exceed the constrans. {δ 1, δ 2, δ s } s a sets of shp-drafts for each shp, and {h 1, h 2, h p } s a sets of deep-sea for each port. F 1, F whch F s a set of fuel Phase 2: Phase 3: Phase 4: Phase 5: f P port. Startng wth route r 1,n=1 of the cluster group R ps Check any port n the route cluster R ps If the any port of the route s exceed restrans the deep-sea and shp-draft allowed, mpose a penalty α. And return to phase 2 for the next route n the same route cluster R ps (r 1,n=n+1 ). Otherwse go to phase 4 Check any port n the route cluster R ps If the route wthout any fuel port, mpose a penalty β. And Return to phase 2 for the next route cluster (r n=n+1 ). Otherwse calculate the total cost. Fnsh the algorthm untl the entre route checked. 5 Real World Problem The proposed method s appled for the real-world problems. Transportaton problem n PT. PELNI (PT. PELayaran Nasonal Indonesa = the Indonesan Natonal Shppng Company) nterestng to use for testng proposed method, because: 1. Indonesa s an archpelago that ncludes roughly 17,508 slands Indonesa where the total area s 741,052 square mles and wth an area of ocean s 35,908 square mles. The Indonesan archpelago stretches 3181 mles from east to west (Longtude: 97ºE E) and 1094 mles from north to south (Lattude: 6 N - 11ºS). Indonesa needs a system of nter-sland transportaton that can assst n overcomng solaton arsng from geographc dfferences. And as developng country, the seabased transports play an mportant role [19]. 2. PT PELNI operated 26 passenger shps on January 2007 wth dfferences n sze and the speed. The shps wll serve 96 ports that have a dfference of geographc characterstcs [2]. 3. Generally, objectve functon of the concept for fndng a set of shps routes s to mnmze total voyage cost. Then becomes a problem s the real stuaton that PT. PELNI s Indonesan state-owned shp company. PT. PELNI apart from beng based on the proft but must gve prorty to the nterests of the people, n vew of the fact that PT. PELNI as Indonesan natonal shppng company that nvolve n the servce of the voyage servce must could ISBN:

5 become a lnk nter-sland and gve the contrbuton towards the transportaton sea based mode n Indonesa. Therefore the objectve s must to fnd a set of routes gvng mnmal total travel cost wth maxmal total dstance travel [19]. ports that served by routes developed by CAB-Method s 472, and t s more than number port could served of exstng route (318 ports). Increasng number of port vsted wll ncrease the number of passenge who could be served. 6 Result and Analyss Z. Ismal [18] proposes several ways for measurng the qualty of heurstcs and Slver et al. [4] stated that a good heurstc should possess the followng four propertes: 1. Realstc computatonal effort to obtan soluton. 2. The soluton should be close to the optmum on the average,.e., we want good performance on the average. 3. The chance of a very poor soluton (.e., far from the optmum) should be law. 4. The heurstc should be as smple as possble for the user to undetand, preferably explanable n ntutve terms, partcularly f t s to be used manually. Durng the executon for the compute result, the combnaton optmal soluton s recorded n Table 1. Table 1. Exstng Route Compare CAB-Method Fgure 3. Number of port Mnmal cost on routes developed by the CABmethod does not reduce the travel dstance, as show n Fgure 4. All the data can be show n the followng graph n Fgures 2, 3 and 4. Fgure 2 shows the salng cost of exstng route s about $US 4,450,185.49, and wll decreasng to 4,254, wth proposed route by usng CAB-Method. The reducton n the voyage cost s $US 75, Fgure 4. Voyage tme Fgure 2. Voyage cost Fgure 3 shows comparng both of exstng route and route developed by CAB-Method. The total number The total voyage tme to serve all port on the avalable route and satsfyng some constrans as propose by CAB-Method s days. It s more than total voyage tme to serve all ports of the exstng route, about days. Increasng voyage tme wll ncrease tme to serve passenge and t s meanng could be servng more passenge. The results are compared both of exstng route and route developed by CAB-Method; generally speakng, the performance of the routes could be mproved by usng the CAB-Method. CAB-Method able use to fnd mnmal total travel cost and wth maxmal dstance travel. ISBN:

6 6 Concluson From the result presented shows that route made by CAB-Method wll ncrease and mprove performance of PT. PELNI (PT. PELayaran Nasonal Indonesa = the Indonesan Natonal Shppng Company) by save the total cost about $US 75, for serve all port n Indonesa. Total number of ports vsted wll ncrease, 318 port when the routes developed by exstng method and to 472 ports when the route developed by usng CABmethods; that enhance the performance of PT. PELNI as a lnk between slands n Indonesa. Route develop by CAB-Method wll ncrease the total round voyage tme about 4.56 days and decreasng total travel dstance about mles. Increased round voyage tme wll be effects to the ablty to serve more passenge. From the results showed that CAB-method s one method that exctng to be studed ts ablty for solve routng problem and encourage further applcatons to other problems. References: [1] D. Ronen, Edtoral to the feature ssue on water transportaton. European Journal of Operatonal Research 55(2), 123, [2] Drektorat Jenderal Perhubungan Laut, Data Dstrbus Angkutan Laut Penumpang Dalam Neger Tahun 2007, Jakarta [3] E. Alba and B. Dorronsoro: A Hybrd Cellular Genetc Algorthm for the Capactated Vehcle Routng Problem, Studes n Computatonal Intellgence (SCI) 82, , [4] E. A. Slver, R. V. Vdal and D de Werra., A tutoral on heurstc methods, European Journal of Operatonal Research, 5: pp , [5] Frank W. Takes Walter A. Koste. Applyng Monte Carlo Technques to the Capactated Vehcle Routng Problem. Leden Insttute of Advanced Computer Scence, Leden Unvety, The Netherlands, [6] G. B. Dantzg and R.H. Ramser, The Truck Dspatchng Problem. Management Scence 6, [7] G. Laporte. Solvng a Famly of Mult-Depot Vehcle Routng and Locaton-Routng Problems. Transportaton Scence. Vol. 22, August No. 3, [8] G. Laporte, The Vehcle Routng Problem: An overvew of exact and approxmate algorthms. European Journal of Operatonal Research 59: , [9] L. Bodn, B. Golden, A. Assad, and M. Ball, Routng and schedulng of vehcles and crews, The State of the Art: Compute and Operatons Research, 10: pp , [10] Lane, D.E., T. D. Heaver and D. Uyeno, Plannng and schedulng for effcency n lner shppng. Martme Polcy & Management 14(2), , [11] M. Chrstansen, K. Fagerholt, B. Nygreenand and D. Ronen., Martme Transportaton. A Chapter for Handbooks n Operatons Research and Management Scence: Transportaton, C. Barnhart and G. Laporte (Eds.), Northholland, Amsterdam, pp , [12] M. C. Mourao, and L. Amado., Heurstcs Method For a Mxed Capactated Arc Routng Problem: A Refuse Collecton Applcaton. European Journal of Operatonal Research. 160: , [13] M. R. Anderberg, M.R., Cluster Analyss for Applcatons. Academc press, New York [14] P. Toth and D. Vgo, edto. The Vehcle Routng Problem. SIAM, [15] P. Toth, D. Vgo, The Vehcle Routng Problem. Monographs on Dscrete Mathematcs and Applcatons. SIAM, Phladelpha, [16] R.L Bowerman, PH Calama, and GB Hall, The spacefllng curve wth optmal parttonng heurstc for the vehcle routng problem. European Journal of Operatonal Research 76, pp , [17] Roberto Baldacc, Mara Battarra and Danele Vgo. Routng a Heterogeneous Fleet of Vehcles. Techncal Report DEIS OR.INGCE 2007/1. Unvety Bologna, [18] Z. Ismal., Development of Heurstc Methods Based On Genetc Algorthm for Solvng Vehcle Routng Problem. Research Vote No: 74285, Unvet Teknolog Malaysa, [19] -. Study on the Development of Domestc Sea Transportaton and Martme n the Republc of Indonesa, ALMEC Corporaton and Japan Marne Scence Inc, ISBN: