OPTIMAL SCHEDULING MODELS FOR FERRY COMPANIES UNDER ALLIANCES

Transcription

1 Jounal of Maine Science and Technology, Vol. 15, No. 1, pp (2007) 53 OPTIMAL SCHEDULING MODELS FOR FERRY COMPANIES UNDER ALLIANCES Shangyao Yan*, Chia-Hung Chen**, Hsin-Yen Chen*** and Tze-Chiang Lou** Key wods: fey, alliance, scheduling, multiple commodity netwok flow poblem. ABSTRACT Fey companies in Taiwan inceasingly ally themselves with othe fey companies as a means of foming moe complete netwoks, to opeate moe efficiently. The moe complex fey fleet outing and scheduling pocesses ae not only impotant in each company s opeations, but also have a beaing on the alliance. In this eseach, we employ netwok flow techniques to constuct seveal coodinated scheduling models to help solve fo the most satisfactoy schedules fo the allied fey companies. Finally, we pefom a case study based on eal opeating data fom two Taiwan fey companies to evaluate the models. The peliminay esults show that the model could be useful fo this type of situation. INTRODUCTION Pape Submitted 03/08/06, Accepted 05/12/06. Autho fo Coespondence: Shangyao Yan. t320002@cc.ncu.edu.tw. *Pofesso, Depatment of Civil Engineeing, National Cental Univesity, Chungli, Taiwan 32054, R.O.C. **Ph.D. candidate, Depatment of Civil Engineeing, National Cental Univesity, Chungli, Taiwan 32054, R.O.C. ***Undegaduate, Depatment of Civil Engineeing, National Cental Univesity, Chungli, Taiwan 32054, R.O.C. In ecent yeas, touism has developed apidly in Taiwan. Fo example, the mass apid tansit system has allowed Danshui to develop into a popula amusement aea. To enhance this development, the govenment has ecently pomoted an Inland Rive Blue Highway plan, aimed at poducing specialized tous, in ode to offe moe sevice options to local touists. Given these cicumstance, the fey companies have stiven to impove thei opeations. Of all the factos, fey fleet outing/scheduling has been the most impotant focus of fey companies, because this not only affects fey boat usage efficiency, the establishment of the timetable, and maintenance and cew scheduling, but is also essential to a fey company s pofitability and its level of sevice. Most fey companies in Danshui cuently use a tial-and-eo pocess fo fey fleet outing and scheduling pactices. The plannes adjust the dafted timetable and the fey fleet outes/schedules by consideing the numbe of available fey boats, thei aveage opeating speed, the tun-aound time at the pots, the fey fleet balance at each pot, and the elated cost/evenue of fey boat movements between pots. Note that the timetable is typically designed by expeience, in accodance with the pojected demand, the maket shae, and the given ight of wateway. This pocess is iteated manually, without optimization fom a systemic pespective. Afte adjustments, the schedule is then sent to be checked fo fey boat maintenance and cew scheduling, with possible mino evisions. Such an appoach is less efficient when sevice netwoks gow, and could possibly esult in an infeio solution. Much eseach has aleady been devoted to ship outing and shipment scheduling poblems, by the maine industy as well as in academic fields [3, 4, 7-9, 13-15]. Ou suvey of the above eseach on shipment scheduling indicates that the focus has been on single caie cago tanspotation, which is fundamentally diffeent fom Taiwan s Inland Rive passenge tanspotation. Passenges in paticula ae moe time sensitive than cago. Moeove, shipment scheduling belongs to the long-haul type of tanspotation, which is fundamentally diffeent. In addition, the fast gowth of the Inland Rive maket in this aea, foces the fey companies to place stategic emphasis on the gowth of Inland Rive tanspotation, to impove thei opeating pefomance, and to povide bette sevices. Actually, thee ae hundeds of diffeent types of alliances in the tanspotation industies, anging fom ageements fo specific outes to full meges. The setting of a good coodinated fey schedule can not only enhance the opeating pefomance of the allied fey companies, but can also act as a useful efeence fo fey company alliance decision-making. Fo example, if it is difficult to develop new OD sevice fey tips, then two fey companies can ente into a complementay alliance, linking existing patial tips to fom a new moe complete complementay tip and to offe new OD sevices. An effective coodinated scheduling model would help these companies find the most

2 54 Jounal of Maine Science and Technology, Vol. 15, No. 1 (2007) satisfactoy fey fleet outes and timetables. At this point in time, howeve, fey companies tying to coodinate alliance schedules, geneally use a tial-and-eo pocess fo fey fleet outing and scheduling fomulation. Schedules ae ecipocally iteated, constucted and evaluated, manually and independently, without optimization fom a systemic pespective. Only afte all this, do the allies check to see whethe thei schedules ae mutually suitable. If they ae not, then each schedule must be modified futhe. The pocess is epeated until satisfactoy esults ae obtained. It is easy to see that such an appoach is neithe effective no efficient, especially when the fey netwok is lage. Infeio solutions can be the esult. How to simultaneously detemine a good coodinated fey schedule that satisfies each company in the alliance is difficult. Recently much academic eseach has been devoted to the solving of alliance poblems [2, 5, 6, 10-12,16, 21]. We have not found any eseach in this liteatue that coves both fey fleet outing and scheduling unde alliances. In addition, the liteatue main focus has been on ai tanspotation, which makes it difficult to apply to the fey alliance scheduling poblem, and to integate fey alliances, fey fleet outing and timetable setting, in shot-tem opeations. In this eseach, we develop seveal coodinated scheduling models that combine fey company alliances, fey fleet outing and timetable setting to help the paticipating fey companies solve fo the most satisfactoy fey fleet outes and timetables fo shot-tem opeations unde paallel and complementay alliances, given the pojected OD demand, fey fleet size, and elated cost data. The poposed models ae expected to be good planning tools fo the allied fey companies. If the given inputs ae changed, o the obtained esults do not eflect the inputs, then the models can be suitably modified and eun until satisfactoy esults ae acquied. Fo simplicity, we focus pimaily on single-fey fleet outing and scheduling but the models could be extended to multi-fey fleet opeations. Futhemoe, fo simplicity of modeling, two fey companies (Fey companies A and B) ae used as examples. Although the scheduling pocess is, in pactice, elated to the fey boat maintenance and the cew scheduling pocesses, these pocesses ae caied out afte fey fleet outing and scheduling. In paticula, maintenance and cew constaints ae athe flexible in pactice, fo the fey companies studied in this eseach. In a few cases the outes and schedules may have to be slightly modified to meet maintenance o cew scheduling issues. To facilitate poblem solving, we theefoe exclude these constaints in ou modeling. The est of this pape is oganized as follows: Fist, we intoduce the models. Then, numeical tests ae pefomed to evaluate the models. Finally, we offe some conclusions. MODELING APPROACH Taditionally, fey companies have used daft timetables as an essential medium fo fey fleet outing and timetable setting. This not only involves too much subjective judgment in the pocess, but also eveals the difficulty of systematically managing the inteelation between supply and demand. To impove the difficulty of dafting multiple-stop timetables fo passenge tanspot, Yan and Tseng [18] suggested using a netwok model, containing fleet flow netwoks and passenge flow netwoks. The model can diectly manage the inteelationships between passenge tip demands and flight supplies, so as to moe effectively assist caies scheduling. Such an appoach has ecently been applied to othe poblems [1, 17]. Refeing to Yan and Tseng [18], a time-space netwok technique is applied to constuct seveal coodinated fey fleet outing and scheduling models fo the pupose of maximizing the fey companies total pofit. Fo ease of modeling, a basic model is ceated fist, followed by seveal stategic models. The majo elements in the modeling, including the fey-flow netwoks, the passenge-flow netwoks, and the mathematical fomulation, ae as follows. 1. Fey-flow netwoks A netwok, as shown in Figues 1 o 2, fo singlefey fleet outing is established fo each company, within the specified time peiod (one day in this study) and location. Note that fo ease of eading, the thin dotted aows indicate the ally s fleet-flow netwok but such dotted aows do not eally exist in the fey-flow netwok. The hoizontal axis epesents the pot locations; the vetical axis stands fo the time duation. All available pots ae included. The two majo components in the netwok ae nodes and acs. Each node epesents a specific pot and a specific time, while each ac epesents an activity fo a fey boat, such as a sevice, a bething peiod, o an ovenight stay. The ac flows expess the flow of fey fleet in the netwok. Thee types of acs ae defined below. (1) Sevice ac A sevice ac epesents a tip connecting two diffeent pots. Diffeent sevice ac densities may be set fo diffeent peiods o pot pais, fo peak/ non-peak hous o the demands fo each pot pai. Fo example, in cuent pactices the designated tip fequency is five minutes fo peak peiods and one fey

3 S. Yan et al.: Optimal Scheduling Models fo Fey Companies unde Alliances 55 Pot-1 Pot-2 Pot-3 Pot-4 Pot-5 Pot-k-1 Pot-k (3) 09:00 09:05 09:10 09:15 (1) 09:20 09:25 09:30 09:35 (2) 09:40 09:45 09:50 Pot-1 Pot-2 Pot-3 Pot-4 Pot-5 Pot-k-1 Pot-k (2) (3) (1) 09:00 09:05 09:10 09:15 09:20 09:25 09:30 09:35 09:40 09:45 09:50 (1) Sevice ac (2) Bething ac (3) Cycle ac 19:45 19:50 19:55 20:00 (1) Sevice ac (2) Bething ac (3) Cycle ac 19:45 19:50 19:55 20:00 Fig. 1. Fey-flow netwok of Fey A. Fig. 2. Fey-flow netwok of Fey B. tip pe half hou fo non-peak hous. Each sevice ac contains infomation about the depatue time, the depatue pot, the aival time, the aival pot and the opeating cost. The time block fo a fey tip is calculated as fom the time when the fey boat is pepaed fo this tip to the time when this tip is finished. Basically, it includes an investigation time pio to depatue, time fo fuelling, passenge embakation and debakation, and the actual tip time. The ac cost is thus the fey boat opeating cost. The ac flow is a binay vaiable denoting the numbe of fey boats (0 o 1) that seve the associated fey tip. The ac flow s uppe bound is one, meaning that a tip can be seved at most once. The ac flow s lowe bound is zeo, implying that no boat seves this tip. In addition, the depatue inteval at the same pot is adjustable to meet each company s opeating equiements. To design sevice acs fo paallel/complementay alliances, the following two points should also be consideed. A. Paallel alliance A paallel alliance efes to the collaboation of two fey companies with the same outes in thei netwoks. In a paallel alliance, the companies togethe sevice a tip. Typically one company chages a fee to its ally fo seving thei passenges. This means that a side constaint should be set fo the tip, ensuing that the ac flows associated with the same sevice ac in the two netwoks ae at most one. As shown in Figue 1, the thick aows (seved by Fey company A) and the thin dotted aows (seved by Fey company B) epesent tips connecting pot 3 and pot 4. Similaly, as shown in Figue 2, sevice acs ae designed to connect pot 4 and pot 3. The two types of tips have the same oigin and destination but ae sepaately distibuted. B. Complementay alliance A complementay alliance efes to a situation whee two fey companies have linked thei existing patial netwoks to fom a new moe complete complementay netwok. In othe wods, taffic is fed to each othe. In a complementay alliance, the companies both fom a fey tip to seve a new OD demand. As shown in Figue 1, the thick aows (seved by Fey company A) and the thin dotted aows (seved by Fey company B) epesent the tips connecting pot 6 and pot k. Fey company A is fist (pot 6 to pot k-1) and Fey company B is second (pot k-1 to pot k), o Fey company B is fist (pot k to pot k-1) and Fey company A is second (pot k-1 to pot 6). Similaly, the sevice acs in Figue 2 ae designed to connect pot 6 and pot k. (2) Bething ac A bething ac epesents the holding of fey

4 56 Jounal of Maine Science and Technology, Vol. 15, No. 1 (2007) boats at a pot in a time window. The ac cost denotes the holding expenses incued. The ac flow s uppe bound is the anchoage capacity (o infinity, if the capacity is lage). This indicates the maximum numbe of fey boats that can be held at this pot duing a specific time window. The ac flow s lowe bound is zeo, implying that no fey boat is held at this pot in this time window. (3) Cycle ac A cycle ac epesents the continuity between two consecutive planning peiods. It connects the end of one peiod to the beginning of the next peiod fo each pot. Such a technique ensues that fleet outes and schedules otate egulaly, which makes them easie to implement in pactice. Note that, without cycle acs, outes may not otate egulaly and deadheading of fey boats could occu, which would incease the opeating cost. Simila appoaches have been applied in othe scheduling poblems [19]. The ac cost is the cost of holding a fey boat ovenight, and is simila to the bething ac cost but with the addition of an ovenight chage. The uppe bound and lowe bound of the ac flow ae set the same as those of the bething acs. 2. Passenge-flow netwoks The time-space netwok technique is applied to model passenge movement at cetain times and locations, fo each fey company. Each passengeflow netwok epesents a specific OD pai fom the oigin-destination table (known as the OD table). A set of passenge-flow netwoks associated with the ODs ae constucted fo each fey company. In paticula, thee types of passenge-flow netwoks, with a simila stuctue, ae ceated fo each fey company, including individual, paallel and complementay alliance netwoks, as shown in Figues 3, 4 and 5, espectively. An individual passenge-flow netwok plans the tanspotation of an OD s passenges, by only its own fey/ fey tips in tems of time and space. A paallel alliance passenge-flow netwok plans the tanspotation of an OD s passenges, by its and its allied fey company s fey/fey tips, but on the same oute. A complementay alliance passenge-flow netwok plans the tanspotation of a new OD s passenges, via its fey tips and its allied fey company s fey tips, in each existing patial netwok, to fom a new complementay netwok. To facilitate poblem solving these netwoks ae designed to be symmetical to the fey-flow netwoks. Since the netwoks fo Fey companies A Pot-1 Pot-2 Pot-3 Pot-4 Pot-5 Pot-6 Pot-k-1 Pot-k 09:00 09:05 09:10 (3) 09:15 (1) 09:20 09:25 09:30 (2) 09:35 09:40 09:45 09:50 Pot-1 Pot-2 Pot-3 Pot-4 Pot-5 Pot-6 Pot-k-1 Pot-k (2) ) (1) (3) 09:00 09:05 09:10 09:15 09:20 09:25 09:30 09:35 09:40 09:45 09:50 (1) Tanspot ac (2) Stay ac (3) Demand ac 19:45 19:50 19:55 20:00 (1) Tanspot ac (2) Stay ac (3) Demand ac 19:45 19:50 19:55 20:00 Fig. 3. Individual passenge-flow netwok fo Fey company A (OD pai: 1->2) Fig. 4. Paallel alliance passenge-flow netwok fo Fey company A (OD pai: 3->4)

5 S. Yan et al.: Optimal Scheduling Models fo Fey Companies unde Alliances 57 Pot-1 Pot-2 Pot-3 Pot-4 Pot-5 Pot-6 Pot-k-1 Pot-k Fig. 5. Complementay alliance passenge-flow netwok (OD pai: 6->k) and B ae simila, to save space, we only show the netwoks fo Fey company A. The hoizontal and vetical axes ae the same as those in the fey-flow netwoks. Hee, a node also epesents a pot at a specific time, but an ac designates an activity showing passenge movement. Thee ae thee types of acs in each type of passenge-flow netwok. They ae defined below. (1) Tanspot ac (1) Tanspot ac (2) Stay ac (3) Demand ac A tanspot ac epesents the tanspotation of passenges fom one pot to anothe. The tip is seved by eithe the oiginal fey company o its ally. The tanspotation time is the same as the coesponding time block fo the associated fey tip in the fey-flow netwok. The ac flow s uppe bound is the fey boat capacity (with pehaps a planning load facto), meaning that the maximum flow in the ac is the loading capacity. The ac flow s lowe bound is zeo, indicating that no passenge fom the coesponding OD is deliveed on the associated fey tip. In addition to the above common chaacteistics, thee ae attibutes specific to each type of passenge-flow netwok, descibed as follows: The ac cost fo an individual passenge-flow netwok, as shown in Figue 3, is a vaiable cost fo (2) (1) (3) 09:00 09:05 09:10 09:15 09:20 09:25 09:30 09:35 09:40 09:45 09:50 19:45 19:50 19:55 20:00 seving a passenge. In a paallel alliance passengeflow netwok, as shown in Figue 4, the associated tanspot ac cost fo a fey tip seved by the individual fey company, is a vaiable cost fo seving a passenge; when the tip is seved by the allied fey company, the associated tanspot ac cost is the cost that the fey company has to pay its ally (usually negotiated between the two fey companies), to compensate fo the tanspot of a passenge. The ac costs fo a complementay alliance passenge-flow netwok, as shown in Figue 5, ae set simila to the ac costs of a paallel passenge-flow netwok. (2) Stay ac A stay ac denotes the holding of a passenge at a pot in a time window. A holding cost (o penalty) is the ac cost fo the time window. Howeve, if the ac just happens to connect eithe the depatue o the aival pot of this netwok s coesponding OD pai, the ac cost is then zeo, because, in pactice, the staying of passenges at such pots is usually not consideed a scheduling decision. The ac cost is adjustable. The ac flow s uppe bound is the pot s passenge sevice capacity, within the netwok s minimum time inteval, implying that the maximum numbe of passenges can be accommodated at this pot in the time window. The ac flow s lowe bound is zeo, meaning that no passenge fom the coesponding OD stays at the pot duing this time window. (3) Demand ac A demand ac connects the aival pot to the depatue pot of this netwok s coesponding OD pai fo a specific time inteval (accoding to the studied fey company, this is 30 minutes fo peak hous and one hou fo non-peak hous). It denotes the sevice demands fo the OD pai fo the time inteval that would actually be seved in the netwok, whethe by the oiginal fey company o its ally. The ac cost is the negative value of the aveage associated ticket fae. The ac flow s uppe bound is the pojected demand fo this OD pai. The aim of the model is to maximize the pofit, which means that not all passenges fo this OD pai will necessaily be seved. The ac flow s lowe bound is zeo, meaning that none of the OD pai s passenges ae seved. Accoding to Yan and Tseng [18], the tip demand fo a specific OD pai can be flexibly divided into seveal demand acs, based on the actual demand distibution, maket chaacteistics, and fey company consideations. In paticula, a demand ac denotes the time inteval within which passenges will wait fo tanspotation, so thee will be no significant loss. In

6 58 Jounal of Maine Science and Technology, Vol. 15, No. 1 (2007) othe wods, the pojected demand fo the time inteval is insensitive to the associated time length. This time inteval, which is detemined by the caie, is based on pactical expeience/judgment, and can be eset. The model can then be eun until satisfactoy esults ae obtained. Note that demand ac time intevals fo peak/ non-peak hous and fo diffeent ODs can be diffeent. Such a design moe effectively models passenge tanspot plans, paticulaly fo multi-stop tips, than does the taditional timetable dafting appoach. The effectiveness depends on the setting of the demand acs by the caie. Finally, if the model esults ae expected to have an impact on the oiginal demand, the inputs can be changed and the model eun, until satisfactoy esults ae acquied. 3. Notations of symbols used in the model fomulation Befoe intoducing the model fomulation, we fist list the symbol notations that will be used in the model fomulation., q :the th and the q th allied fey companies, espectively; R : the set of all allied fey companies. In this eseach, R = {, q}; n : the n th OD pai; o : the o th individual OD pai; a : the a th paallel OD pai; m : the m th complementay OD pai; N : the set of all ODs fo the th allied fey company; PN : the set of all paallel alliance passenge-flow netwoks fo the th allied fey company; CPN : the set of all complementay alliance passenge-flow netwoks fo all allied fey companies; IPN : the set of the individual passenge-flow netwoks fo the th allied fey company; A, NF : the set of all acs and nodes in the fey-flow netwok fo the th allied fey company; CF : the set of all cycle acs in the fey-flow netwok fo the th allied fey company; B n, NP n : the set of all acs and nodes in the n th passenge-flow netwok fo the th allied fey company; AF : the numbe of available fey boats used to povide sevices fo the th allied fey company; FF : the set of all sevice acs in the fey-flow netwok fo the th allied fey company; IFF : the set of all sevice acs in the fey-flow netwok fo the th allied fey company, associated with the fey tips seved by the fey company itself; PFF : the set of all sevice acs, assocaited with paallel alliance, in the fey-flow netwok fo the th allied fey company; PPFA a : the set of the delivey acs in the a th paallel alliance passenge-flow netwok fo the th allied fey company; CPFA m : the set of the delivey acs in the m th complementay alliance passenge-flow netwok; IPFA o : the set of the delivey acs in the o th individual passenge-flow netwok fo the th allied fey company; PPDA a : the set of the demand acs in the a th paallel alliance passenge-flow netwok fo the th allied fey company; CPDA m : the set of the demand acs in the m th complementay alliance passenge-flow netwok; IPDA o : the set of the demand acs in the o th individual passenge-flow netwok fo the th allied fey company; K : the fey boat capacity (pehaps with a planning load facto); FUij : the ac(i,j) flow s uppe bound in the feyflow netwok fo the th allied fey company; PUij n : the ac(i,j) flow s uppe bound in the n th passenge-flow netwok fo the th allied fey company; : the ac(i,j) cost in the fey-flow netwok fo C ij T n ij VC n ij CVC n ij PPDC a ij D m ij the th allied fey company; : the ac(i,j) cost in the n th passenge-flow netwok fo the th allied fey company; : the delivey ac(i,j) cost in the n th passengeflow netwok fo the th allied fey company, associated with a fey boat seved by the fey company itself, which is a vaiable cost incued by the th allied fey company fo deliveing a passenge; : the delivey ac(i,j) cost in the m th complementay passenge-flow netwok, associated with the fey boat seved by the fey company itself and its ally, which is a vaiable cost incued by the fey companies fo deliveing a passenge; : the delivey ac(i,j) cost in the a th paallel alliance passenge-flow netwok fo the th allied fey company, which is the cost that the th allied fey company pays to the othe allied fey company fo compensating its delivey of a passenge; : the pojected passenge tip demand associated with the demand ac (i,j) in the m th complementay alliance passenge-flow netwok. Note that the demand is seved by complementay tips of both fey companies;

7 S. Yan et al.: Optimal Scheduling Models fo Fey Companies unde Alliances 59 PT f ij p n ij : the popotion of pofit (evenue minus vaiable cost fo seving passenges) shaed by the th allied fey company unde a complementay alliance, which is detemined in advance though an ageement between both fey companies. Note that the sum of all the popotions is equal to 1, namely PT = 1. : the ac(i,j) flow in the fey-flow netwok fo the th allied fey company; : the ac(i,j) flow in the n th passenge-flow netwok fo the th allied fey companies; 4. Basic model fomulation Besides the fey-flow and the passenge-flow netwoks intoduced above, thee ae seveal issues that need to be consideed in the modeling: (1) the numbe of equied fey boats in the netwok should not exceed the numbe of available fey boats, (2) the numbe of passenges tanspoted on a fey tip should not exceed the seving fey boat s capacity, (3) the passenge demand seved by both fey companies should not exceed the pojected one, and (4) given a paallel alliance the same sevice ac can be seved at most once in both fey-flow netwoks. Theefoe, fou coesponding types of side constaints ae designed duing the poblem fomulation: (1) the sum of the cycle ac flows in each fey-flow netwok should not be geate than the numbe of available fey boats, (2) the sum of all tanspot ac flows coesponding to the same sevice ac should not exceed the sum of each sevice ac flow multiplied by the fey boat capacity, (3) each demand ac flow in the complementay alliance passenge-flow netwoks should be less than o equal to the associated pojected passenge demand and (4) the sum of all the ac flows in the two paallel alliance fey-flow netwoks, coesponding to the same fey tip, should be less than o equal to one. Based on the fey-flow and the passenge-flow netwoks, as well as the side constaints, we fomulate the model as a mixed intege netwok flow poblem. The objective of this model is to flow the fey boats and passenges simultaneously in all netwoks at a minimum cost. Since the ticket evenue fom the passenge-flow netwoks is in the fom of a negative cost, this objective is equivalent to the maximization of pofit. The model is fomulated as follows: Minimize Z = T n ij Subject to C f ij + n p n ij ij A ij ij B n (1) f ij f j NF ki = 0 k NF i NF, R (2) n p ij n =0 k NP n j NP n p ki i NP n, n N, R (3) f ij ij CF AF R (4) n p ij m + p n F ij m CPN Kf ij ij IFF, R (5) n p ij nq + p n N ij n N q Kf ij ij FF \IFF, R (6) p m ij D m ij ij CPDA m, m CPN (7) f ij 1 ij PFF (8) 0 f ij FU ij ij A, R (9) 0 p n ij PU n ij ij B n, n N, R (10) f ij I ij A, R (11) The model is fomulated as a mixed intege multiple commodity netwok flow poblem, in which the objective function, Eq. (1), is to minimize the total system cost of the allied fey companies. Constaints (2) and (3) ensue flow consevation at evey node in each fey/passenge-flow netwok. Constaint (4) denotes that the numbe of fey boats used in each feyflow netwok should not exceed the available numbe of fey boats. Constaint (5) keeps the passenge tanspot volume, including the oveall numbe of individual and complementay passenges seved, within the fey boat s caying capacity fo the fey tips seved by the fey company itself. Eq. (6) keeps the passenge tanspot volume within the fey boat s caying capacity fo the fey tips seving both a fey company s and its ally s passenges. Note that R = {, q}. Eq. (7) indicates

8 60 Jounal of Maine Science and Technology, Vol. 15, No. 1 (2007) that, each demand ac flow in the complementay alliance passenge-flow netwoks should be less than o equal to the associated pojected passenge demand. Constaint (8) indicates that unde the paallel alliance, the allied fey companies should simultaneously povide at most one fey tip. Constaints (9) and (10) hold all the ac flows within thei bounds. Eq. (11) ensues the integality of the fey flows. It should be noted that the caies could slightly adjust the individual model solutions, by a post optimization analysis, to bette match eal equiements. Also note that fo ease of compaing the pefomance befoe and afte they enteed into alliances, we need to calculate thei individual opeating pofit (which is equal to the evenue minus the cost) fo shot tem opeations, fo each allied fey company. Namely the Eq. (1) could be subdivided fo two allied fey companies. Suppose that z, F cost and F ev epesent the th allied fey company s pofit, cost and evenue, espectively. The cost and evenue ae divided into thee pats, paallel netwoks, complementay netwoks, and individual netwoks, as shown in Eqs. (12) and (13), espectively. In Eq. (12), the fist tem is the fey flow cost, the second is a vaiable passenge seving cost paid by the fey company itself, unde a paallel alliance, the thid is a vaiable cost fo seving its ally s passenges, unde a paallel alliance, the fouth is the cost fo the fey company pays to its ally to compensate fo the delivey of passenges, unde a paallel alliance, the fifth is a vaiable cost fo seving passenges, unde a complementay alliance, and the last is a vaiable cost fo seving passenges paid by the fey company itself, in its own individual netwoks. F cos t = C ij f ij + ij A + a PN + a PN ij PPFA a PN q ij PPFA aq a PN ij PPFA a + PT + VC a aq ij p ij PPDC a a ij p ij m CPN ij CPFA m VC o o ij p ij CVC ij m p ij m VC a a ij p ij o IPN ij IPFA o (12) In Eq. (13), the fist tem is the evenue eceived by the fey company itself fo seving passenges, unde a paallel alliance, the second is the evenue eceived fom its ally fo seving the allied fey company passenges, unde a paallel alliance, the thid is the evenue fo seving passenges, unde a complementay alliance, and the last is the evenue eceived by the fey company fo seving passenges in its own individual netwoks. Note that, since the evenue fom the passenge-flow netwoks is in the fom of a negative cost, a minus sign to the fou tems is added to tansfe the value into positive evenue. F ev = a PN ij PPDA a a PN PT T a a ij p ij PN q ij PPFA aq m CPN ij CPDA m T o o ij p ij PPDC aq aq ij p ij T ij m p ij m o IPN ij IPDA o (13) 5. Stategic model fomulations In this eseach paallel and complementay alliances ae modeled simultaneously. Howeve, in pactice, the alliance type is dependent upon the natue of the allied outes. In othe wods, paallel and complementay alliances may not be simultaneous in all cases. Howeve, the basic model can be suitably modified to confom to eal pactices. In paticula, the basic model can be simplified to model eithe paallel o complementay alliance along. The detailed modeling of the above two modifications ae now descibed: (1) Paallel alliance Each fey company s complementay alliance passenge-flow netwoks ae emoved. The sevice acs in both fey-flow netwoks not seving any demand ae emoved. The fey boat loading constaint (5) is modified as n N n p ij Kf ij, ij IFF, R, to emove all passenge flows in the complementay passenge-flow netwoks. Constaint (7) is emoved. Note that it is assumed that additional tips of both fey companies ae constucted in the complementay netwok. Without the complementay alliance, these tips would not be seved by eithe fey company. The esulting model is thus a paallel alliance scheduling model. (2) Complementay alliance The paallel alliance passenge-flow netwoks ae modified by the emoval of tanspot acs seved by the allied fey company to become individual passengeflow netwoks. The associated OD demands should then be modified (in geneal, they ae educed), fo each fey company. The fey boat loading constaint (6) is emoved. The bundle constaint (8) is emoved. Afte these modifications, we constuct a complementay alliance scheduling model.

9 S. Yan et al.: Optimal Scheduling Models fo Fey Companies unde Alliances Model Applications Although the models could be applied in many diffeent situations, to save space, two examples ae addessed below: (1) Feasibility of alliance Geneally, the fey companies use the maximum pofit fo thei opeation objective. If each fey company does not allow its shot-tem opeating pofit less than a taget value afte alliance, then an additional constaint should be added into the basic model fo this concen. Let Z be the negative value of the th allied fey company s pofit afte alliance, and PF be the negative value of the th allied fey company s taget pofit. The additional constaint is as follows: Z = C ij f ij + T n n ij A n ij p ij PF R ij B n (14) Note that the model could be infeasible with too lage taget pofits fo the allied fey companies. That is, with the coodinated scheduling optimization it still cannot help the allied fey companies achieve thei taget pofits. Unde this cicumstance, the fey companies should adjust thei taget pofits and esolve thei schedule to obtain a esult satisfactoy to both. Duing the pocess, each allied fey company can also examine the alliance effect to itself that would help adjust its alliance decision in the futue. (2) Minimum numbe of tips The above models wee designed without specifying a minimum numbe of fey tips fo any OD pai. Howeve, if the numbe of tips fo some OD pais in the coodinated schedule decease enough to cause the level of sevice and the associated demand to dop, then the constaint fo a minimum numbe of tips fo the associated OD pais can be intoduced. Fo example, let OF st be the numbe of the diect tips fo the st th OD pai of the th allied fey company befoe they ente into an alliance. Also let A st be the set of all diect sevice acs associated with the the st th OD pai in the th fey-flow. Then, the additional constaints fo each allied fey company may now be added as follows: ij A st f ij OF st fo some st N, R (15) NUMERICAL TESTS To test how well the models may be applied in the eal wold, we pefomed numeical tests using opeating data fom two Taiwan fey companies, with easonable assumptions. We used the C compute language, coupled with the mathematical pogamming solve, CPLEX 9.0, to build the model and to solve the poblems. The tests wee pefomed on a Pentium 4 2.0G with 1. 5Gb of RAM in the envionment of Micosoft Windows XP. We fist used the opeating data to build the model, and then solved the poblems. Finally, we demonstated seveal examples fo how to apply the models. 1. Data analysis and Test esults The numeical tests wee mainly based on the weekday data obtained fom two Taiwan fey company opeations (Fey companies A and B) duing Septembe of 2005, with easonable simplifications. Five pots wee seved by Fey companies A and B, each with a single fleet of 8 fey boats. Each fey boat has a caying capacity of 95 seats. To peliminaily evaluate the model pefomance, the pojected OD demand (fo individual, paallel and complementay OD pais) was estimated with easonable assumptions. In paticula, fo individual OD pais and paallel OD pais, the pojected OD demands mainly efeed to the elated epots of the two fey companies. Fo complementay OD pais, the pojected OD demands, which wee newly developed, wee diectly set based on planning staff expeiences of the two fey companies. All the cost paametes and othe inputs, such as the tip time, the distance between two pots, and the bething time, ae pimaily based on actual opeating data, with easonable simplifications. Moeove, fo the paallel alliance, the cost that a fey company must pay to its allies, to compensate fo the tanspot of a passenge, is set as ninety pecent of the ticket fae. Fo the complementay alliance, accoding to eal pactice, the popotion of the pofit is set to be 0.5 fo both fey companies (i.e. PT = 0.5). Note that the fixed cost (i.e., the sunk cost) and the indiect cost (fo example, the capital investment, depeciation, maintenance o ental chages), which ae constant in shot-tem opeations, ae not included in the model. In othe wods, the pofit calculated hee is the shot-tem opeating pofit, athe than the actual oveall pofit of the system. Howeve, the optimization of the shot-tem pofit can incease the long-tem system pofit. In addition, to peliminaily evaluate the poposed models, the cuent schedules of the two fey companies (paallel alliance) obtained by the tial-andeo expeience-based method ae used. These esults obtained ae efeed to as actual opeations. In

10 62 Jounal of Maine Science and Technology, Vol. 15, No. 1 (2007) accodance with eal pactices, the esults obtained by the paallel alliance model, with the same pojected demand as fo actual opeations, ae compaed with that of the actual opeations. As shown in Table 1, the objective value (OBJ) of the paallel alliance was bette than that of actual opeations by 1.07%, showing that fom a systematic optimization pespective the paallel alliance model is supeio to the cuent tial-and-eo expeience-based method. Moeove, compaed with the individual fey company s objective values of actual opeations, using the paallel alliance model, the objective value of Fey company A inceased fom to (an impovement of 1.22%), and the objective value of Fey company B inceased fom to (an impovement of 0.94%). In addition, with the use of the paallel alliances model the fequency (tips/day) fo both fey companies inceased (fom 227 to 231 fo A and fom 242 to 245 fo B). Similaly, the sevice ates fo both fey companies inceased (fom 94.7% to 97.4% fo A and fom 96.2% to 97.9% fo B). The aveage load factos fo both fey companies also inceased (fom 83.5% to 84.69% fo A and fom 84.9% to 85.57% fo B). All of the esults show that the paallel alliance model could impove ove the cuent tial-andeo method used in actual opeations. In addition to the above compaison, we also compaed the esults of diffeent alliance type opeations with those of the individual opeations. Note that when the pojected OD demand fo a paallel OD pai is sepaated into two individual ones, then accoding to staff expeience the demand will be suitably educed. The esults ae shown in Table 2. Note that the tem mixed alliance denotes the combination of both paallel and complementay alliances. The best solution was yielded by the mixed alliance, with an objective value of The paallel alliance was next, with an objective value of , followed by the complementay alliance with an objective value of The individual opeations pefomed most pooly, with an objective value of ( and fo Fey companies A and B). Fom the above esults, we found that the mixed alliance integated not only the paallel outes but also connected the complementay outes, and was theefoe bette than the paallel alliance and the complementay one, espectively. In addition, the paallel alliance is moe effective than the complementay one. It is also Table 1. Test esults fo actual opeations and paallel alliance Actual opeations Paallel alliance Fey Fey Fey Fey company A company B company A company B OBJ (NT$) Computation time (sec) NA Fey size Fequency (tips/day) Sevice ate (%) Aveage load facto (%) Table 2. Test esults Individual opeations Paallel alliance Complementay alliance Mix alliance Fey Fey Fey Fey Fey Fey Fey Fey company A company B company A company B company A company B company A company B OBJ (NT$) Computation time (sec) Fey size Fequency (tips/day) Sevice ate (%) Aveage load facto (%)

11 S. Yan et al.: Optimal Scheduling Models fo Fey Companies unde Alliances 63 found that compaed with the individual fey company s objective values, with a mixed alliance, the objective value of Fey company A inceased fom to (an impovement of 1.55%), and the objective value of Fey company B inceased fom to (an impovement of 1.86%). Fo a paallel alliance, the objective value of Fey company A inceased fom to (an impovement of 1.32%), and the objective value of Fey company B inceased fom to (an impovement of 1.21%). Fo a complementay alliance, the objective value of Fey company A inceased fom to (an impovement of 0.44%), and the objective value of Fey company B inceased fom to (an impovement of 0.405%). The above esults show these types of alliance do impove fey company opeations. In Table 2, the available fey boats of both fey companies (8 of A and 8 of B) ae all used up. In addition, the fequency (tips/day) fo both fey companies inceased fo the complementay, the paallel and the mixed alliances, espectively (fo A fom 226 to 230, 231 and 235; and fo B fom 241 to 244, 245 and 248). Similaly, the sevice ates fo both fey companies inceased, fo the complementay, paallel and mixed alliances (fo A fom 94.9% to 97.1%, 97.4% and 97.6%, espectively; and fo B fom 96.3% to 97.8%, 97.9% and 98.2%, espectively). The aveage load factos fo both fey companies, fo the complementay, the paallel and the mixed alliance, also inceased (fo A fom 83.6% to 84.23%, 84.69% and 85.08%, espectively; and fo B fom 84.96% to 85.27%, 85.57% and 86.12%, espectively). The above esults showed that the opeations of two fey companies wee all impoved, iegadless of the alliance types. In paticula, alliances though complicated combinatoial optimization, especially the mixed type of alliance, can lead to the effective integation of fey fleet outes and schedules and hence the impovement of the fey company opeations. Finally, the fey fleet flows obtained above could not yet be diectly put into pactice without identifying each fey boat path in the fey-flow netwoks. The flow decomposition method [19] was applied to tace the path of each fey boat. An example of the fey boat outes is shown in Figue 6, the full lines epesent a fey company A s fey boat oute and the dotted lines epesent a fey company B s fey boat oute. 2. Model Applications/Scenaio analyses Fo bevity, we use the mixed alliance model to demonstate some model applications. Futhemoe, to evaluate the model pefomance fo solving middle/ :00 9:00 9:15 9:30 9:45 10:00 10:00 10:30 11:15 11:30 13:00 16:45 17:30 17:45 18:00 18:45 17:00 19:15 19:30 20:00 11:45 12:00 Fig. 6. An example of fey boat outes. lage scale poblems, we also pefomed a scenaio analysis. (1) Feasibility of alliance 10:45 12:30 12:30 12:45 17:15 20:00 17:00 17:30 17:45 18:15 10:45 12:15 18:30 18:45 19:30 In pactice, the fey companies may conside the alliance feasibility to themselves. To demonstate the applications, we solved the poblem incopoating the alliance feasibility of taget pofit. Assume the taget pofit fo each fey company afte the alliance to be times of that without the alliance (i.e., PF 1 = 1.015*( ) and PF 2 =1.015*( )). We added the associated constaint with the alliance feasibility into the model and then solved the poblem. As shown in Table 3, the total objective value slightly inceased fom to (an incease of 0.009%), due to the additional constaint. Both fey companies eached thei taget pofits, with (= *( ) <1.015*( )) and

12 64 Jounal of Maine Science and Technology, Vol. 15, No. 1 (2007) (= *( ) <1.015* ( )), espectively, implying that this application can help the allied fey companies make suitable schedules fo achieving thei taget pofits. (2) Minimum numbe of tips In this scenaio, we assume that afte the alliance the numbe of tips fo each fey company deceased fo one individual OD pai, causing the levels of sevice to dop. It is detemined to maintain a minimum numbe of tips (fo example, consideing the long tem opeation stategy), equal to thei oiginal numbe befoe the alliance fo these two individual OD pais (i.e., 22 tips pe day fo Fey company A and 23 tips pe day fo Fey company B). Theefoe, two moe side constaints ae added, as in Eq. (13). We eun the model with the othe inputs emaining the same. The esults ae shown in Table 4. Although the esults show that the pofits of Fey companies A and B deceased by and 25.86, espectively, the levels of sevices wee maintained. Note that Fey company A emoves 1 tip and Fey company B 1 tip, because, fo the system optimization pespective unde limited esouces, inceasing the numbe of tips fo the two OD pais would decease the numbe of tips fo all othe OD pais. (3) Poblem scales To evaluate the model pefomance fo middle/ lage scale poblems, we tested 3 moe poblem instances with diffeent scales anging fom 2 to 4 times the oiginal scale. Fo each poblem instance, we added a numbe of pots, fey boats and OD pais to ou oiginal poblem. The distance, tip times and ticket faes fo each new pot-pai, the bething time at each new pot, and othe cost paametes wee andomly set in elation to the oiginal poblem. Based on the oiginal OD demands, as well as the oiginal/new fey fleet size, we andomly set the tip demand fo each OD pai. In paticula, the tip demand fo each OD pai was suitably inceased with the fey fleet size. The paametes, CPLEX 9.0 and the test envionment wee set the same as in the pevious test. Table 5 shows the test esults fo diffeent poblem instances. As the poblem scales inceased, the computation time Table 3. Test esults fo the alliance feasibility Fey company A Mix alliances Fey company B OBJ (NT$) GAP (%) Computation time (sec) Fey size 8 8 Fequency (tips/day) Sevice ate (%) Aveage load facto (%) Table 4. Test esults fo minimum numbe of tips Fey company A Mix alliances Fey company B OBJ (NT$) GAP (%) Computation time (sec) Fey size 8 8 Fequency (tips/day) Sevice ate (%) Aveage load facto (%) Table 5. Test esults fo poblem scales Oiginal 2 times 3 times 4 times (5 pots) (10 pots) (15 pots) (20 pots) OBJ (NT$) GAP (%) Computation time (sec) OD pais Fey fleet size Fequency (tips/day) Sevice ate (%) Aveage load facto (%)

13 S. Yan et al.: Optimal Scheduling Models fo Fey Companies unde Alliances 65 inceased. Fo example, a mixed alliance with 20 pots, 48 OD pais and 32 fey boats fo each fey company equied a computation time of seconds (3.58 times the oiginal one). This shows the model s efficiency fo lage-scale poblems. To save space, the eade may contact the authos fo the detailed esults that ae not discussed hee. CONCLUSIONS In this eseach, we develop seveal coodinated scheduling models designed to help the paticipating fey companies solve fo the most satisfactoy fey fleet outes and timetables unde the alliance. It is expected that the models will be useful planning tools by which the allied fey companies can detemine the most suitable fey fleet outes and timetables fo shottem opeations. We employ netwok flow techniques to constuct the models. Each model includes multiple passenge- and fey-flow netwoks that can fomulate the flows of passenges and allied fey boats in the dimensions of time and space. A numbe of side constaints ae set accoding to eal opeating equiements. The models ae fomulated as multiple commodity netwok flow poblems which ae solved using a mathematical pogamming solve. Numeical tests utilizing the domestic opeations of two fey companies in Danshui wee pefomed to peliminaily evaluate the models. The esults show that the coodinated type of alliance not only educes the opeating cost but also inceases pofit. The mixed type of alliance out-pefoms both the paallel and complementay alliance. To demonstate how the model can be applied, we also outlined seveal scenaios with diffeent conditions. Although the peliminay test esults show that the models ae potentially useful fo scheduling, especially fo domestic fey companies, moe tests o case studies should be conducted, so that caies may gasp thei limitations, befoe putting them to pactical use. The models, the test esults, and the model applications, should all be useful efeence mateial fo allied fey companies to detemine the most satisfactoy shot-tem fey fleet outes and schedules. The models may also be suitably modified fo alliances with moe than two fey companies. The extension of a two-fey company alliance to a multifey company alliance, and the incopoation of othe objectives, opeating constaints o alliance stategies involved in actual opeations, could be diections fo futue eseach. Note that although the technique fo designing demand acs used in this eseach is an impovement ove the taditional timetable dafting appoach, the numbe of passenges tanspoted may not be accuately calculated, due to the complicated passenge chaacteistics and choice behavios that occu in the eal maket. If the povided sevices (including the tip fequency and the tip tavel time) do not eflect the pojected demand, the poposed models can be eun, with suitable demand modification, until satisfactoy esults ae acquied. Howeve, if the povided sevices vay significantly with the demand o due to passenge chaacteistics and choice behavios, then this technique might be difficult to apply in pactice. The poposed models can howeve be suitably modified by incopoating a passenge choice model and a genealized passenge flow netwok, as was done in Yan et al. [20]. This could also be a diection fo futue eseach. ACKNOWLEDGEMENTS This eseach was suppoted by a gant fom the National Science Council of Taiwan. We thank the two fey companies fo poviding the test data as well as thei valuable opinions. REFERENCES 1. Banhat, C., Knike, T., and Lohatepanont, M., Itineay-Based Ailine Fleet Assignment, Tanspotation Science, Vol. 36, No. 2, pp (2002). 2. Bueckne, J.K. and Whalen, W.T., The Pice Effects of Intenational Ailine Alliances, Jounal of Law and Economics, Vol. 43, No. 2, pp (2000). 3. Cho, S.C. and Peakis, A.N., Optimal Line Fleet Routing Stategies, Maitime Policy and Management, Vol. 23, No. 3, pp (1996). 4. Chistiansen, M. and Fageholt, K., Robust Ship Scheduling with Multiple Time Windows, Naval Reseach Logistics, Vol. 49, No. 6, pp (2002). 5. Dennis, N., Scheduling Issues and Netwok Stategies fo Intenational Ailine Alliances, Jounal of Ai Tanspot Management, Vol. 6, pp (2000). 6. Evans, N., Collaboative Stategy: An Analysis of the Changing Wold of Intenational Ailine Alliances, Touism Management, Vol. 22, pp (2001). 7. Fageholt, K., Ship Scheduling with Soft Time Windows An Optimisation Based Appoach, Euopean Jounal of Opeational Reseach, Vol. 131, pp (2001). 8. Jaamillo, D.I. and Peakis, A.N., Fleet Deployment Optimisation fo Line Shipping Pat 2: Implementation and Results, Maitime Policy and Management, Vol. 18, pp (1991). 9. Lane, D.E., Heave, T.D., and Uyeno, D., Planning and Scheduling fo Efficiency in Line Shipping, Maitime Policy and Management, Vol. 14, pp (1987). 10. Li, M.Z.F., Distinct Featues of Lasting and Nonlasting Ailine Alliances, Jounal of Ai Tanspot