Capacity Reservation for Time-Sensitive Service Providers: An Application in Seaport Management

Transcription

1 Capacty Reservaton for Tme-Senstve Servce Provders: An Applcaton n Seaport Management L. Jeff Hong Department of Industral Engneerng and Logstcs Management The Hong Kong Unversty of Scence and Technology Clear Water Bay, Hong Kong, Chna Xaowe Xu Department of Supply Chan Management and Marketng Scences Rutgers, The State Unversty of New Jersey Newark, New Jersey 070 Shenghao Zhang 3 Department of Industral Engneerng School of Management, X an Jaotong Unversty X an, 70049, Chna Emal: hongl@ust.hk Emal: xaowex@andromeda.rutgers.edu 3 Emal: shhzhang@mal.xjtu.edu.cn

2 Capacty Reservaton for Tme-Senstve Servce Provders: An Applcaton n Seaport Management Abstract Ths paper studes a capacty management problem n whch a faclty provder offers ts faclty to two servce provders. The faclty provder can ether pool the servce provders together to share the faclty or reserve a dedcated faclty for the servce provders. The servce provders determne ther servce capacty levels to serve each market wth lnear tme-senstve demand. We assume that both the faclty provder and the servce provders maxmze demand rates. We fnd that the faclty provder s optmal capacty strategy crtcally depends on the rato of the servce provders demand loss rates, whch are the demand loss per unt tme ncrease. The faclty provder prefers the reservaton strategy to the poolng strategy f one servce provder s demand loss rate s sgnfcantly larger than the other s. Otherwse, the faclty provder prefers the poolng strategy. From the servce provders perspectve, the servce provder who has a large demand loss rate prefers the reservaton strategy, but the servce provder who has a small demand loss rate prefers the poolng strategy. We also study a centralzed system, n whch the faclty provder fully controls the servce provders and hence s able to elmnate the negatve effect of the faclty competton between the servce provders. We fnd that poolng s always optmal for the faclty provder, whch suggests that faclty capacty competton s a prerequste condton for not poolng the servce provders together. We connect these manageral nsghts wth the operatons of seaports n Eastern Asan area e.g., Hong Kong, Shangha and etc. Keywords: capacty game, capacty management, port management, tme-based competton.

3 Introducton The world s economes are becomng more nterrelated as a result of ncreasng nternatonal tradng and globalzaton. Worldwde merchandse exports have been growng at an average rate of 5.5% per year snce 000 and the total exports and mports across the world were over $7.5 trllon n 007 The World Trade Organzaton 008. Chna has emerged as a manufacturng engne for the globe. Its exports were $.43 trllon approxmately 3% of GDP and mports were $.3 trllon approxmately 6% of GDP n 008 Natonal Bureau of Statstcs of Chna 009. Seaports play an ncreasngly mportant role to connect the world and support the globalzaton trend. The world s port throughput has had an annual growth rate of 0% snce 000 The World Bank 007, Module, Page 49 and the largest ten contaner ports n the world, among whch sx of them are located n Chna Shangha, Hong Kong, Shenzhen, Guangzhou, Nngbo-Zhoushan and Qngdao, handled 7 mllon TEUs twenty-foot equvalent unts n 008 Hong Kong Marne Department 009a. Seaports provde facltes for carrers to dock ther vessels and load/unload ther cargos, and they have sgnfcant mpacts on the operatonal effcency of the carrers. The port servce ndustry reported about $50-55 bllon n busness recepts n 006 and ths amount s expected to grow to $75-80 bllon n 009 The World Bank 007, Module, Page 7. The total captal nvestment requred to buld a modern contaner port s often enormous. For nstance, the Port of Tanjung Pelepas n Malaysa, whch started operatng n 999, requred a captal nvestment of $745 mllon The World Bank 007, Module, Page 8. Capacty expanson of an exstng port s also expensve. The estmated cost to redevelop the exstng Rajv Gandh Contaner Termnal n Inda and buld a new termnal nearby was about $460 mllon n 004 The World Bank 007, Module 3, Page 73. To recover the enormous captal nvestment, a port has to handle a large volume of vessels and cargos to acheve economes of scale. For port owners and operators, an mportant objectve s to keep port facltes busy and to use them effcently. But a busy port may cause long and unpredctable tme delays for carrers. For nstance, the average contanershp watng tme at the Port of Cartagena n Colomba s about two hours and when the berth occupancy rate a measure of port utlzaton s 50%, but the tme jumps to 0 days when the berth occupancy rate ncreases to 90% The World Bank 007, Module, Page. Long tme delay n a port often causes a loss to carrers as ther customers are usually senstve to the tme spent on the transportaton route. To reduce the tme delay and avod competng wth other carrers for port facltes, carrers are nclned to havng dedcated port facltes.e., facltes that can only be used by the desgnated carrer. By usng dedcated facltes, carrers gan more control over port operatons and ensure better servce qualty than by sharng facltes wth other carrers. A strategc problem for a port authorty s to decde whether to pool all carrers together to

4 share the port facltes or to allocate the facltes to ndvdual carrers. When a port pools the vessels from all carrers together and fully utlzes ts facltes, ths poolng effect generally leads to more effcent use of the facltes. In contrast, the port partally loses the poolng effect f t reserves part of ts facltes for a specfc carrer, snce dedcated facltes cannot be used to handle other carrers vessels even when they are dle. However, the poolng strategy s not perfect for the port. When carrers are put together, they may compete for the port facltes by ncreasng the vessel frequency n order to provde better servce for ther customers. Ths competton effect may result n congeston and offset the beneft of the poolng effect. The competton effect exsts n logstc systems n whch servce users share publc facltes. For nstance, the traffc jam at major U.S. arports n the summer of 007 was partally caused by the ncrease n flght frequences of U.S. arlne companes, who ntended to offer convenent schedules for travelers Busness Week 007. Snce usng dedcated facltes separates the operatons of dfferent carrers, ths reservaton strategy elmnates the competton effect as well as the poolng effect. Hence, t s mportant for both a port authorty and carrers to understand the tradeoffs between poolng and reservaton strateges and the nteractons among ther capacty decsons. In ths paper, we consder a faclty provder e.g., a destnaton seaport that offers ts faclty to two servce provders e.g., carrers, who shp cargos from two dfferent orgn ports to the same destnaton port. The demand rate at each orgn port s measured by the number of vessels requred to shp cargos out per unt tme. Each servce provder decdes ts servce capacty the shppng frequency of vessels on ts route, whch s also the demand rate of the servce provder at the destnaton port. The faclty s capacty s measured as the number of vessels capably handled per unt tme. The faclty provder chooses ether to let the servce provders share the faclty by the poolng strategy or to allocate a dedcated faclty to the servce provders and serve them separately by the reservaton strategy. The demand rate of the servce and the shppng frequency determne the tme that cargo spends at the orgn port. Smlarly, the shppng frequency and the faclty capacty determne the tme that vessels spend at the destnaton port of the route. We assume that the demand rate of the servce decreases lnearly n the total processng tme spent at the orgn and destnaton ports. We also assume that both the faclty provder and the servce provders maxmze ther demand rates. Both assumptons are consstent wth the practce n the martme ndustry. It has long been known that shppng n a tmely fashon s an mportant busness am n transportaton ndustres. Due to large captal nvestment, achevng economes of scale and hgh capacty utlzaton s crtcal for the success of both ports and carrers. Ths makes cargo volume one of the most mportant performance measures n the ndustry see, e.g., Stopford 997 and The World Bank 007, Module 3, Page 85. The focus of our study s on dentfyng condtons under whch the faclty provder and servce 3

5 provders should adopt the poolng/reservaton strategy and optmze ther capacty decsons. We consder three scenaros. In the frst scenaro, the faclty provder adopts the poolng strategy. The servce provders determne ther servce capacty levels and compete for faclty usage. We fnd a unque Nash equlbrum for ths scenaro. In the second scenaro, the faclty provder allocates faclty capacty to each servce provder, who determnes the servce capacty gven ts dedcated faclty capacty. Fnally, n the thrd scenaro, we study the frst-best outcome of a centralzed system, n whch a central planner jontly chooses the faclty capacty management strategy and the servce capacty levels. We obtan closed-form solutons for all three scenaros. We fnd that the faclty provder s optmal choce between the poolng strategy and the reservaton strategy crtcally depends on the rato of the demand loss rates of the servce provders. The demand loss rate s defned as the demand loss f the total processng tme ncreases by one unt. It s proportonal to the servce provder s potental market sze.e., the potental volume of cargos on a route and the tme senstvty of demand. The faclty provder prefers the reservaton strategy f one servce provder s demand loss rate s much larger than the other s e.g., the demand loss rato s greater than 4 n Theorem 5; otherwse, t prefers the poolng strategy. Our results are consstent wth ndustry practce. For nstance, the Maersk Lne accounts for almost 80%-90% of the traffc at the Port of Salalah and domnates the rest of small carrers at the port. Hence, t s not surprsng that the port provdes Maersk wth dedcated facltes. However, at the ports of Sngapore, Hong Kong and Shangha, whch are the three largest ports n the world, none of the carrers has such a clear sze domnance over other carrers. Hence, these ports do not provde dedcated facltes The World Bank 007, Module 3, Page 86. We also fnd that f one servce provder s demand loss rate s four tmes larger than the other s, the domnant servce provder prefers the reservaton strategy Theorem 7. Ths s why large carrers often request dedcated facltes. But ther requests may be dened by a port authorty, who only grants dedcated facltes f the demand loss rato s larger than 4 Theorem 5. In contrast, for a servce provder, whose demand loss rate s less than the other s, the poolng strategy s preferred. Fnally, we study a centralzed system, n whch the faclty provder fully controls the servce provders and hence s able to elmnate the negatve effect of the faclty competton between the servce provders. We fnd that poolng s always optmal for the faclty provder, whch suggests that faclty capacty competton s a prerequste condton for not poolng the servce provders. The rest of the paper s organzed as follows. In Secton, we provde a bref revew of the related topcs n the lterature. The model s ntroduced n Secton 3. Then, we study the poolng strategy, the reservaton strategy and the centralzed system n Sectons 4, 5 and 6, respectvely. We make comparsons between the poolng and reservaton strateges n Secton 7 and draw conclusons n Secton 8. All proofs are ncluded n the Appendx. 4

6 Lterature Revew Speed has become an mportant strategc tool for frms to gan market share Stalk and Hout 990 and Blackburn 99. For servce and logstc companes, fast delvery s a key to attractng customers. Servce speed has been wdely studed n the operatons management lterature. Kala et al. 99 studed a duopoly game n whch two frms compete for market shares by choosng ndvdual server capacty levels n a queueng system. L 99 studed an olgopoly game n whch frms stock up nventores to cut delvery tme and attract tme-senstve customers. L and Lee 994 studed prce competton n a tme-senstve market and found that a frm wth hgh processng capacty enjoys a prce premum. Lederer and L 997 developed a competton model n whch customers have dfferent senstvty to delvery tme and frms can dfferentate themselves by prcng, producton and schedulng polces. So 000 studed a prce and delvery tme compettve game n whch frms satsfy customer demand wthn a guaranteed delvery perod at a prefxed probablty level. He found that hgh capacty frms offer better tme guarantees than do low capacty frms and an ncrease of tme senstvty n customer demand strengthens ths dfferentaton. Boyac and Ray 003 studed a product dfferentaton problem n whch a frm determnes the prces of a regular and an express product and the delvery tme of the express product. They examned the relatonshp among capacty cost, tme dfferentaton and prce dfferentaton. Allon and Federgruen 008 studed prce and watng tme competton wth general queueng systems and examned how the compettve behavor of servce provders changes wth the queueng system characterstcs. Chayet and Hopp 007 studed a sequental entry game n whch frms compete on capacty, prce and lead tme. They found that the entrant needs superor operatonal capablty to overcome the ncumbent s frst-mover advantage. The exstng delvery tme lterature focuses on a smple servce system, n whch a servce provder can delver the servce by herself and hence s always able to shorten the delvery tme and mprove servce qualty by ncreasng her servce capacty. However, n practce, many servce systems nvolve wth multple partes and ncreasng the servce capacty of one servce provder may not help mprovng the overall servce system performance. For nstance, a martme system ncludes carrers and port authortes. Snce a seaport has a lmted capacty n processng vessels, a carrer may not be able to shorten the cargo delvery tme when pushng the vessel frequency close to the port s handlng capacty. Smlar phenomena occur n many logstc systems, n whch servce provders share logstc assets e.g., ports, roads and other publc facltes to provde servce to customers and gan economes of scales Fuller et al However, when traffc s heavy n these systems, publc logstc facltes become congested, whch causes delays and hurts servce qualty. Unlke the prevous lterature, ths paper studes the mpacts of both servce and publc faclty 5

7 capacty levels on the delvery tme. Moreover, the bottleneck of publc logstc facltes becomes severe when multple servce provders compete on a fxed amount of faclty capacty. It s well known that a user of a publc resource often gnores the negatve externalty that she/he mposes on other users e.g., Havv and Rtov 998. Ths gnorance s the cause of the congeston n many logstc systems as n the arport congeston case mentoned n Secton. One way to solve ths ssue s to use ncentve-compatble prcng schemes see, e.g., Mendelson and Whang 990 and Ha 998, whch are wdely adopted by publc transportaton authortes. Another way s to allocate dedcated facltes to certan types of users, whch s commonly practced by port authortes and s the focus of ths paper. It s well know that combnng separate subsystems nto one system may mprove the overall system effcency, snce the combnaton reduces the chance of dleness of subsystems and generates economes of scale see, e.g., Smth and Whtt 98 and Whtt 999. However, f customers have heterogenous characterstcs, then mergng queues may be counterproductve. For nstance, f customers fall nto classes wth dfferent servce tme dstrbutons, then keepng dfferent types of customers nto separate queues may be optmal Smth and Whtt 98 and Whtt 999. Yu et al. 009 studed a capacty management problem, n whch a set of ndependent frms can share capacty. They found that capacty poolng may not be optmal f the workloads of frms are sgnfcantly dfferent. Another reason of not mergng queues s that customers have dfferent tme senstvty parameters. Pangburn and Stavrulak 008 studed a jont prcng and capacty management problem and found that capacty poolng s suboptmal f customers are heterogenous n ther tme senstvty. See other reasons of not mergng queues n Rothkopf and Rech 987. In our paper, the poolng decson for the faclty provder not only depends on the characterstcs of the servce provders.e., the demand loss rate but also whether or not there exsts competton for faclty usage. In the decentralzed case.e., the faclty provder does not control the servce provders, f the characterstcs of the servce provders are sgnfcantly dfferent, then the poolng strategy s not optmal for the faclty provder. Ths s because competng for faclty usage between the servce provders creates congeston and delays, whch offset the poolng beneft. In contrast, n the centralzed system.e., the faclty provder fully controls the servce provders, the poolng strategy s always optmal for the faclty provder, snce there s no faclty capacty competton. Hence, one of the mportant reasons of not poolng self-nterested servce provders s the negatve effect of faclty capacty competton. 3 The Model In ths secton, we frst ntroduce a general framework for the port system n our problem. Second, we propose a stylzed model that s analytcally tractable. Fnally, we summarze a three-month nvestgaton at Kwa Tsng Contaner Termnals of Hong Kong, whch provdes some emprcal 6

8 Fgure : The martme system. observatons to support the stylzed model. 3. The Problem Framework We consder a faclty provder that provdes facltes to servce provders. For nstance, a port provdes berths to carrers, who shp or transshp cargos to and from the port. The faclty provder has a total faclty capacty of K, whch s measured as the number of vessels that the faclty can handle per unt tme. Throughout the paper, we assume that the total faclty capacty s fxed, snce t s often very tme-consumng and/or expensve to change the faclty capacty. For smplcty, we consder only two servce provders. Ths s suffcent to demonstrate the tradeoff between the capacty reservaton and poolng strateges. We assume that the servce provders serve dfferent markets that do not nteract wth each other. For nstance, carrers may serve dfferent routes, whch have dfferent orgns but share the same destnaton port see Fgure. In ths case, the demands for the carrer s servces rarely affect each other. We let λ be the demand rate for servce provder, whch s measured as the number of vessels needed to shp the cargo out of the orgn port per unt tme, and assume that the demand rate strctly decreases n the total delvery tme of servce. The servce provders determne ther servce capacty levels e.g., the frequency of vessels travelng on the route. We denote the servce capacty of servce provder as µ, where =,. In the martme ndustry, the total transportaton tme of cargos can be decomposed nto three phases: the dwell tme t d for the next avalable vessel at the orgn port, the shppng tme on the ocean, and the faclty tme t f that a vessel spends watng for and usng port facltes at the destnaton port. Snce the second phase of the total transportaton tme s ndependent of the servce and faclty capacty levels, we let the total processng tme t = t d + t f and assume that the demand rate λ = D t s decreasng n t. In the dwell phase, only when both the faclty and vessels of servce provders are avalable at the orgn port, cargos can be loaded and shpped out. To focus on the nteracton between the 7

9 destnaton port and carrers capacty decsons, we assume that the faclty at the orgn port s capable of handlng cargos and always avalable whenever there s a vessel arrved. Ths s lkely to occur f the orgn port has a dedcated faclty for servce provders. Hence, the dwell tme for servce provder s a functon of the demand rate λ and servce capacty µ, whch s denoted as t d λ, µ. We assume that t d λ, µ s ncreasng n the demand rate λ, but decreasng n servce capacty µ. The dwell tme t d λ, µ should go to nfnty as λ and µ get close to each other.e., there s no suffcent servce capacty to handle the demand. In the faclty phase, the faclty tme s a functon of the faclty capacty and the total amount of vessel traffc that uses the faclty. If servce provders share the faclty, the faclty tme functon s t f µ +µ, K. If servce provder has dedcated faclty capacty K, the faclty tme s t f µ, K. The faclty tme functon s expected to decrease n the faclty capacty level, but ncrease n the amount of vessel traffc. The faclty tme should go to nfnty as the amount of vessel traffc approaches to the faclty capacty level. As a justfcaton of these propertes, we consder the example of the Port of Cartagena n Colomba The World Bank 007, Module, Page. In 993, the average berth occupancy rate the utlzaton of the faclty capacty was 90% and the contanershp watng and turnaround tme the faclty tme was about 3 days. The port ncreased ts capacty sgnfcantly snce then. In 003, the average berth occupancy rate was 50% and the contanershp watng and turnaround tme was reduced to less than 9 hours. Durng the same perod, due to the sgnfcant ncrease n vessel frequency servce capacty, the cargo dwell tme was also reduced from more than 30 days to only days for cargo shpped to other ports. By the monotonc propertes of the dwell and faclty tmes, the total processng tme t s expected to decrease n the faclty capacty level, but ncrease n the demand rate. The total processng tme t should go to nfnty as the servce capacty level the frequency of vessels approaches to ether the demand rate or the faclty capacty level. Hence, the total processng tme should be a U-shaped curve of the servce capacty level. These generc propertes should hold n martme systems. Notce that the total processng tme and demand rate nteract wth each other. When servce provders share the faclty, the equlbrum demand rate λ s determned by λ = D t d λ, µ + t f µ + µ, K. Snce D t s decreasng n t and t d λ, µ s ncreasng n λ, Equaton determnes a unque demand rate functon λ µ, µ, K gven µ, µ and K. Smlarly, f servce provder has a dedcated faclty K at the destnaton port, the demand rate functon λ µ, K s the soluton of λ = D t d λ, µ + t f µ, K. Due to the large captal nvestment needed to buld ports and purchase contanershps, the total 8

10 cargo volume s a prmary concern of both ports and carrers. For nstance, the World Bank ponted out that the major concern of carrers s ther market share whle ther port handlng charges are of secondary mportance The World Bank 007, Module 3, Page 85. To a port authorty, the overall contrbuton of the port operatons to the local economy, whch s correlated wth the cargo volume, s often more mportant than ts own proftablty. Hence, we assume that the faclty provder and the servce provders maxmze ther demand rates. When the poolng strategy s adopted, each servce provder s capacty decson affects the other s total processng tme and hence demand rate. We can model the strategc nteracton between servce provders capacty decsons as a smultaneous game and look for the Nash equlbrum of the game. When the reservaton strategy s adopted by the faclty provder, there s no strategc nteracton between servce provders capacty decsons. The faclty provder decdes how to allocate ts total capacty K to two servce provders wth the objectve of maxmzng the total demand rate. The preference over the poolng and reservaton strateges can be determned by comparng all partes performance under the two faclty management strateges. The problem framework we descrbed s generc. By pluggng n the exact form of the total processng tme and demand functons, the tradeoffs between poolng and reservaton strateges can be made at least va a smulaton-based approach. The exact form of the total processng tme s determned by the operatonal detals of the martme system as shown n Fgure, such as, the operatonal polces of the orgn and destnaton ports and carrers. Gven the complexty of the martme system, t s extremely cumbersome, f possble, to derve a closed-form of the total processng tme. Snce the poolng decson s strategc for the faclty provder and has longterm mpact on all partes n the martme system, we avod to specfy the detaled operatonal nformaton of the martme system, whch tends to change over tme. Instead, we choose to use a stylzed model of the total processng tme, whch s mathematcally tractable and enables us to gan manageral nsghts. These manageral nsghts can be formed as rules of thumb for strategc seaport capacty management. 3. The Stylzed Model We assume that the total processng tme has a smple formula, that s, t = K µ + µ λ, where K s the faclty capacty avalable to servce provder. Notce that K represents the dedcated faclty capacty for servce f the faclty provder adopts the reservaton strategy, or K = K µ f the faclty provder adopts the poolng strategy. The above formula satsfes all generc propertes of the tme functons ntroduced n Secton 3. and hence captures the strategc mpact of the faclty and servce capacty levels on the delvery tme. Moreover, we provde some emprcal evdence to support ths specfc form of the total processng tme functon n Secton 3.3. We assume a lnear tme-dependent demand functon, that s, Dt = A θ t, where A 9

11 s the potental market sze and θ s the tme senstvty parameter. Notce that we normalze the shppng tme on the ocean as zero ths can be done by adjustng the parameters of the demand functon. The lnear demand form s wdely adopted n the lterature see, e.g., Kala et al. 99 and So 000. Wth the exact form of the total processng tme and demand functons, we have the demand rate, as a functon of K and µ, A λ K, µ = θ + µ K µ where δ = A A θ K µ µ, f K K 4K θ < µ < K > 4θ ; otherwse, λ K, µ = 0. See dervaton detals n the Appendx. δ + 4A θ ], K + K 4K θ We solve the servce provder s demand maxmzaton problem as shown n Proposton. Proposton Assume that K > 4θ.. The optmal servce capacty s µ K = 3K +A 4 4 K A + 6A θ ;. The optmal demand rate s λ K = λ K, µ K = K +A K A + 6A θ ; 3. µ K > λ K > 0; 4. The optmal servce capacty µ K and the demand rate λ K are ncreasng n K and A, but decreasng n θ. By Proposton, as the avalable faclty capacty K and/or the market sze A ncrease, servce provder ncreases ts servce capacty to attract customer demand. But as customers become more tme senstve.e., an ncrease of θ, the demand for the servce drops, whch reduces the requred servce capacty. The condton of K > 4θ n Proposton mples that customers are not extremely tme senstve and/or the avalable faclty capacty s moderately large. An nvaldaton of ths condton shuts down the operatons of servce provder.e., µ K = λ K = 0. Snce ths s a trval case, we wll avod t n the rest of ths paper. 3.3 Emprcal Observatons We conducted a three-month nvestgaton from October 9, 008 to January 4, 009 on the vessel arrval and processng pattern at Kwa Tsng Contaner Termnals of Hong Kong, whch has been the thrd busest contaner port n the world snce 007, just after Sngapore and Shangha. The data source s from the Vessel Traffc Management System Reports publshed by Hong Kong Marne Department Hong Kong Marne Department 009b. These reports enable us to trace key tme ponts of a vessel n and out of Kwa Tsng Contaner Termnals. In these reports, there are four key tme varables: ETA Estmated Tme of Arrval, ATA Actual Tme of Arrval, ETD Estmated Tme of Departure and ATD Actual Tme of Departure. There are 636 vessel records wth complete nformaton about the four key tme varables durng the nvestgaton perod. ETA was announced approxmately three days before the arrval of vessels. We let DT A = AT A ET A be the tme dfference between the actual arrval tme and estmated arrval tme. 0 and

12 Fgure : The hstogram of P T. Among the 636 vessel records, 4 vessels arrved at least 30 hours before the estmated arrval tme.e., DT A < 30 and 4 vessels arrved at least 30 hours after the estmated arrval tme.e., DT A > 30. The average value of DT A s 5.54 hours and ts standard devaton s 8.69 hours. Ths mples that vessel arrvals are affected by random factors and are not perfectly scheduled. Hence, vessel arrvals should be modeled va a stochastc process e.g., a Posson process. Smlarly, our data mples that the vessel departure tme s nether perfectly scheduled.e., ET D and AT D are dfferent n most vessel records. Hence, the tme that a vessel spends at the port s uncertan. We let P T = AT D AT A, whch s the tme perod that a vessel spends at the port. The mean and standard devaton of P T are.03 hours and 5. hours respectvely. Fgure shows the hstogram of P T. Ths hstogram s rght-skewed and ndcates that the dstrbuton of P T may be an Erlang dstrbuton. In the stylzed model, the total processng tme s composed of the formula of the expected system tme of an M/M/ system. Notce that an exponental dstrbuton s an Erlang- dstrbuton. Hence, our data roughly supports the total processng tme functon n the stylzed model. In Sectons 4-7, we show that ths smple formula of the total processng tme functon results n clean manageral nsghts, whch can be used as rules of thumb for strategc seaport capacty management. 4 The Poolng Strategy When the faclty provder adopts the poolng strategy, the servce provders determne ther ndvdual servce capacty levels and compete on the faclty capacty. The avalable faclty capacty to servce provder s K = K µ, whch depends on the servce capacty of the other servce provder. Ths mples that each servce provder s capacty decson affects the other s capacty decson. We model ths as a smultaneous game and study the pure strategy Nash equlbrum of the game.

13 We let µ POOL and λ POOL respectvely denote the equlbrum servce capacty level and demand rate of servce provder, and we let Λ POOL = λ POOL + λ POOL denote the total demand rate of the faclty provder under the equlbrum. To ensure that both servce provders use the faclty, we make the followng assumpton. Assumpton K maxa, A + 8 maxθ, θ. Ths assumpton mples that the total faclty capacty s larger than the potental market sze of each servce provder. We let µ K be the best response functon of servce provder. By Clam of Proposton and Assumpton, K µ K = K A K A + 6A θ > K A / > 4θ for =,. By Clam 4 of Proposton, µ K µ K for any K K, where K = K µ. Hence, K µ K > 4θ for any K K and the condton of Proposton always holds at any equlbrum. By Clam 3 of Proposton, µ POOL > λ POOL mples that both servce provders use the faclty under the poolng strategy. > 0 for =,. Hence, Assumpton By Clams and of Proposton, the equlbrum servce capacty µ POOL rate λ POOL must satsfy the followng equatons: + A µ POOL = 3 K µ POOL 4 4 K µ λ POOL POOL + A = K µ POOL A + 6A θ, K µ POOL A + 6A θ, and the demand for both =,. By solvng the equatons through varous transformatons, we obtan a unque pure strategy Nash equlbrum n Theorem. Theorem If Assumpton holds, then there exsts a unque pure strategy Nash equlbrum such that both servce provders use the faclty. At the equlbrum, the servce capacty and demand rate of servce provder are respectvely µ POOL A θ = A M A θ + A θ + Ξ POOL + M + M 6 + Ξ POOL M > 0, λ POOL A θ = A M A θ + A θ + Ξ POOL + M > 0, and the total demand rate of the faclty provder s Λ POOL = A + A + K M + Ξ POOL ], where M = A + A K and Ξ POOL = 4A θ + A θ.

14 Wth the closed-form equlbrum n Theorem, we derve the comparatve statcs of servce capacty levels and demand rates as the market condtons and faclty capacty change. Proposton If Assumpton holds, then:. Servce provder s equlbrum servce capacty µ POOL and demand rate λ POOL are ncreasng n K, A and θ, but decreasng n A and θ ;. The faclty provder s equlbrum total demand rate Λ POOL s ncreasng n K, A and A, but decreasng n θ and θ. Proposton shows that an ncrease of faclty capacty K allows the servce provders to ncrease ther servce capacty levels. As a result, the demand rate of each servce provder and the total demand rate ncrease. A servce provder ncreases ts servce capacty to attract customers as ts potental market sze A ncreases. Ths causes a drop n the avalable faclty capacty for the other servce provder and hence a reducton n the other servce provder s capacty and demand rate. However, the total demand rate ncreases. An ncrease of customers tme senstvty θ drves down the customer demand rate and hence the requred servce capacty of servce. Ths ncreases the avalable faclty capacty for the other servce provder, who then ncreases ts servce capacty to attract more customers. 5 The Reservaton Strategy Suppose that the faclty provder decdes to allocate dedcated faclty capacty K to each servce provder. Ths reservaton strategy elmnates not only the gamng behavor of the servce provders but also the poolng effect. Let Λ RES denote the total demand rate for the faclty under ths scenaro. By Proposton, f K > 4θ, then servce provder chooses servce capacty µ K, whch brngs n demand λ K for the faclty provder. If K 4θ, whch mples that the dedcated faclty capacty for servce provder s too small, then servce provder chooses to drop out of ts market and not to use the faclty. In ths case, the dedcated faclty generates no demand and s wasted. Hence, t s not optmal for the faclty provder to allocate any dedcated capacty K 0, 4θ ]. By Proposton, we have Λ RES K, K = { = K + A µ RES We let K RES and λ RES K A + 6A θ, f K > 4θ and K + K = K; K + A K A + 6A θ, f K = K and K = 0. be the optmal dedcated faclty capacty for servce provder and denote as the servce capacty level and demand rate of servce provder, who s allocated faclty capacty K RES. The optmal total demand rate for the faclty provder s Λ RES = Λ RES K RES, K RES. We make the followng assumpton to avod the trval case that the faclty provder causes one servce provder to drop out. 3

15 Assumpton A 6θ, where =,. Ths assumpton mples that customers are not extremely tme senstve and/or the potental market sze s large. Theorem If Assumptons and hold, then the optmal faclty capacty, the servce capacty level of servce provder and the demand rate are respectvely K RES A θ = A A θ + A + A K > 0, A θ µ RES A θ = A 4 A θ + M + Ξ RES A θ + 3M λ RES = A A θ A θ + A θ and the total demand rate of the faclty provder s Λ RES = > 0, M + Ξ RES + M > 0, A + A + K M + Ξ RES ], where Ξ RES = 4 A θ + A θ. Wth the closed-form solutons n Theorem, we derve the comparatve statcs of the servce capacty levels and demand rates as the market condtons and faclty capacty change. Proposton 3 If Assumptons and hold, then:. The optmal dedcated faclty capacty K RES for servce provder, ts servce capacty µ RES and demand rate λ RES are ncreasng n K and A, but decreasng n A ;. If A + A K, then the optmal dedcated faclty capacty K RES for servce provder, ts servce capacty µ RES and demand rate λ RES are ncreasng n θ, but decreasng n θ ; 3. If A +A < K, then the optmal servce capacty µ RES of servce provder and ts demand rate λ RES are decreasng n θ and θ, but the optmal dedcated faclty capacty K RES s ncreasng decreasng n θ θ ; 4. The faclty provder s total demand rate Λ RES s ncreasng n K, A and A, but decreasng n θ and θ. Notce that most comparatve statcs n Proposton 3 are parallel to the ones n Proposton, except the comparatve statcs wth respect to the tme senstvty parameters. Notce that A + A represents the total market sze and A + A > <K means that the faclty provder s capacty s not tght. In case that the faclty provder s capacty s tght.e., A + A > K, as the customer tme-senstvty θ ncreases, servce provder s market becomes relatvely more attractve. Hence, the faclty provder should allocate more faclty capacty to servce provder and decrease the dedcated capacty for servce provder. As more faclty capacty becomes avalable, servce provder ncreases ts servce capacty to attract more customer demand. In 4

16 contrast, servce provder cuts ts servce capacty and loses demand, as ts dedcated faclty capacty s reduced. In case that the faclty provder s capacty s not tght.e., A + A < K, as the customer tme-senstvty θ ncreases, the faclty provder adopts a very dfferent strategy, that s, she moves faclty capacty from servce provder to servce provder and tres to keep the tmesenstve customers of servce provder, even though ths causes servce provder to cut ts servce capacty and lose demand. 6 The Centralzed System We consder a centralzed system n whch the faclty provder not only determnes ts own faclty operatons strategy but also the servce capacty µ to maxmze the total demand rate λ + λ. Ths provdes the frst-best outcome for the overall system performance. Under the centralzed system, the faclty provder can operate ts faclty wth two strateges:. By the reservaton strategy, the faclty provder allocates faclty capacty K to serve only servce provder, where K + K = K;. By the poolng strategy, the faclty provder allows both servce provders to share the entre faclty. Snce the faclty provder determnes the servce capacty level µ, ths elmnates the competton effect and the negatve externalty of the servce provders gamng behavor demonstrated n Secton 4. Hence, the poolng strategy s expected to domnate the reservaton strategy. We establsh ths result n the followng theorem. Theorem 3 It s always optmal for the faclty provder to adopt the poolng strategy under the centralzed system. By Theorem 3, we focus on the poolng strategy. By ths strategy, the faclty provder maxmzes Λ CEN µ, µ = λ K µ, µ + λ K µ, µ, where λ K, µ s defned n Equaton, K = K µ and =,. Theorem 4 If Assumptons and hold, then the optmal servce capacty and demand rate for servce provder under the centralzed system are respectvely µ CEN = A A θ M + M Ξ + Ξ CEN CEN A θ + A θ λ CEN = A A θ + A θ M + Ξ CEN A θ + A θ and the optmal total demand rate of the faclty provder s A + A + K Λ CEN = where Ξ CEN = A θ + A θ + A θ + A θ. M + Ξ CEN + ] A θ M M + Ξ CEN ], > 0, > 0, 5

17 Fgure 3: a. Optmal servce capacty levels µ CEN and µ CEN ; b. Optmal demand rates λ CEN and λ CEN ; c. Optmal total demand rate Λ CEN. Wth the closed-form solutons n Theorem 4, we derve the comparatve statcs of the servce capacty levels and demand rates as the market condtons and faclty capacty change. Proposton 4 If Assumptons and hold, then:. Servce provder s servce capacty µ CEN and demand rate λ CEN are ncreasng n K;. The faclty provder s total demand rate Λ CEN s ncreasng n K, A and A, but decreasng n θ and θ. By Proposton 4, an ncrease n the faclty capacty allows the faclty provder to ncrease the servce capacty levels and demand rates. Although the comparatve statcs of the total demand rate Λ CEN wth respect to the market condtons A and θ are parallel to the ones n Propostons and 3, the optmal servce capacty µ CEN and ndvdual demand rate λ CEN as the market condtons change. Ths s demonstrated n Example. may not be monotonc Example We let K = 650, A = A = 300 and θ = 5 and vary θ 0, 5]. Fgure 3 shows the optmal servce capacty µ CEN, the ndvdual demand rate λ CEN and the total demand rate Λ CEN. As customers become more tme-senstve to servce, the faclty provder ntally ncreases the servce capacty µ CEN to cut the dwell tme, but the eventual demand rate decrease n Fgure 3b reduces the requred servce capacty µ CEN n Fgure 3a. To shorten the faclty tme and the total processng tme of servce, the faclty provder ntally reduces servce capacty µ CEN n Fgure 3a, whch causes the demand rate of servce to drop slghtly n Fgure 3b. But the eventual decrease of servce capacty µ CEN makes more faclty capacty avalable to servce and drves up servce capacty µ CEN and demand rate λ CEN n Fgures 3a and b. As shown n Fgure 3c, the total demand rate Λ CEN s decreasng as the market condtons of servce deterorate, whch s consstent wth Proposton 4. Example demonstrates that the servce capacty decsons become much more complcated regard to the market condton changes f the decsons are jontly made for the best nterest of 6

18 the whole system than f the decsons are separately made for the best nterest of each ndvdual servce provder. 7 Comparsons and Manageral Insghts We compare the three scenaros studed n Sectons 4-6. We let β = A θ for both =,. Notce that β represents the demand loss of servce provder f the total processng tme of servce ncreases by one unt. In the example of port operatons, a carrer wth a large market sze often has a large value of β. Furthermore, we defne γ = β /β as the rato of the two demand loss rates. Notce that γ = /γ. Frst, we study the faclty provder s preference for the poolng and reservaton strateges. 7. The Faclty Provder s Preference We compare the total demand rates of the faclty provder under the three scenaros n Sectons 4-6, whch are respectvely Λ POOL, Λ RES and Λ CEN. The followng theorem ranks them. Theorem 5 If Assumptons and hold, then:. Λ CEN > Λ POOL and Λ CEN > Λ RES ;. When < γ < , Λ RES < Λ POOL ; 3. When γ = 7 ± 4 3, Λ RES = Λ POOL ; 4. When γ < or γ > , Λ RES > Λ POOL. From the faclty provder s vewpont, usng capacty reservaton elmnates the competton effect between the servce provders, but t also elmnates the poolng effect. If the negatve externalty of the servce provders gamng behavor offsets the poolng beneft, the faclty provder should adopt the reservaton strategy. Theorem 5 gves a clear crteron on ths tradeoff, whch only depends on the demand loss rato γ. It s better for the faclty provder to adopt the reservaton strategy f and only f the demand loss rato s sgnfcantly dfferent from,.e., γ > or γ < /4. The condton means that the demand characterstcs of the two servce provders must be sgnfcantly dfferent. When θ and θ are smlar, one servce provder has to be domnant so that the faclty provder adopts the reservaton strategy. Ths result s supported by the real practce n port operatons. As reported by the Port Reform Toolkt The World Bank 007, ports domnated by one carrer often provde dedcated facltes to the domnant carrer, but ports that serve equally szed carrers are unwllng to provde dedcated facltes. For nstance, the Maersk Lne accounts for almost 80%-90% of the traffc at the Port of Salalah, and t s much larger than other carrers usng the port. Hence, t s not surprsng that the port provdes the Maersk Lne wth dedcated facltes. On the other hand, at the ports of Sngapore, Hong Kong and Shangha, whch are the three largest ports n the world, no carrer s domnant over the others. Hence, these ports do not provde dedcated facltes The World Bank 007, Module 3, Page 86. 7

19 When to pool separate subsystems together has been studed n the queueng lterature. Pangburn and Stavrulak 008, Smth and Whtt 98, Whtt 999 and Yu et al. 009 found that poolng s not optmal f customer characterstcs e.g., servce tme dstrbutons and tme senstvty are sgnfcantly dfferent. Ths observaton s confrmed n our model. One of the reasons of not poolng s that the servce provders have sgnfcantly dfferent demand loss rates, whch mples that the poolng beneft s low. However, unlke the prevous lterature, we dentfy another reason of not poolng, that s, poolng self-nterested servce provders ntroduces faclty capacty competton and creates faclty over-utlzaton and congeston. The congeston effect can be measured by the total traffc.e., the total servce capacty level that uses the faclty. Theorem 6 If Assumptons and hold, then µ POOL > µ CEN > µ RES, where µ POOL = µ POOL + µ POOL, µ RES = µ RES + µ RES and µ CEN = µ CEN + µ CEN. Theorem 6 suggests that the faclty s always n the busest mode f the faclty provder pools the servce provders together. On the postve sde, the poolng strategy makes the faclty accessble for both servce provders, whch stmulates the total traffc amount and ncreases the faclty utlzaton rate.e., µ POOL > µ RES and µ CEN > µ RES. However, on the negatve sde, the poolng strategy allows the servce provders to compete on the faclty capacty, whch makes the faclty overused and creates congeston.e., µ POOL > µ CEN. Snce congeston causes delays, lowers servce qualty and hence hurts the demand rates, the negatve effect of the faclty capacty competton may offsets the poolng beneft. Ths s why the faclty provder may not prefer the poolng strategy. Notce that f the faclty provder can elmnate the faclty capacty competton between the servce provders e.g., n the centralzed system, then poolng s always optmal. condton for not poolng the servce provders together. Hence, faclty capacty competton s a prerequste Second, we study the servce provders preferences for the capacty reservaton and poolng strateges. 7. The Servce Provders Preference We compare the demand rates of servce provder under the two scenaros n Sectons 4-5, whch are respectvely λ POOL and λ RES. The followng theorem ranks them. Theorem 7 If Assumptons and hold, then there exsts a threshold γ, 4 such that λ POOL > λ RES f γ < γ and λ POOL < λ RES f γ > γ. By Theorem 7, f the demand loss rato s smaller than, then the servce provder wth the smaller demand loss rate prefers to sharng the faclty capacty wth the other servce provder, snce the beneft of accessng the whole faclty for the smaller servce provder domnates the 8

20 Fgure 4: The preferences for the capacty reservaton and poolng strateges. negatve externalty generated by the larger servce provder. In contrast, f the demand loss rato s larger than 4, then the servce provder wth the larger demand loss rate prefers to usng capacty reservaton. Ths explans why large carrers often ask for dedcated facltes. However, ther requests may be dened by a port authorty f ther market szes are not suffcently domnant.e., the demand loss rato s less than 4 n Theorem 5. Fgure 4 summarzes the faclty provder s and servce provders preferences for the capacty reservaton and poolng strateges under dfferent market condtons. Fnally, we consder a numercal example. Example We let K = 5, A = 00, θ = and θ = 0.5 and vary A 0, 00]. Fgure 5 shows the optmal demand rates Λ I, λ I and λ I, and servce capacty levels µi, µ I and µ I, where I {POOL, RES, CEN}. As shown n Fgure 5a, f the market sze of servce provder s very small.e., A < 30, the optmal total demand rate Λ RES s hgher than Λ POOL, whch mples that the faclty provder prefers the reservaton strategy. Otherwse, Λ RES < Λ POOL, whch mples that the faclty provder prefers the poolng strategy. Ths s consstent wth Theorem 5. Fgure 5b shows that the total traffc rate µ POOL under the faclty competton s the hghest among the three scenaros.e., poolng, reservaton and centralzaton. Ths s consstent wth Theorem 6. Notce that servce provder has a larger market sze and s more tme-senstve than servce provder, whch mples that servce provder has a larger demand loss rate than servce provder. As shown n Fgures 5c and e, servce provder prefers the reservaton strategy snce λ RES > λ POOL, but servce provder prefers the poolng strategy snce λ RES < λ POOL. Ths s consstent wth Theorem 7. Snce servce provder s less tme-senstve than servce provder, the faclty competton and ts congeston consequence has less negatve effect on servce provder than on servce provder. Hence, servce provder behaves more aggressvely under the faclty competton than servce provder. Ths causes that servce provder s optmal servce capacty µ POOL s the hghest among the three scenaros as shown n Fgure 5f, but servce provder s optmal servce capacty µ POOL 9

21 Fgure 5: a. Optmal total demand rate Λ I ; b. Optmal total servce capacty µ I ; c. Optmal demand rate of servce provder λ I ; d. Optmal servce capacty of servce provder µi ; e. Optmal demand rate of servce provder λ I ; f. Optmal servce capacty of servce provder, where I {POOL, RES, CEN}. µ I 0

22 s the lowest among the three scenaros as shown n Fgure 5d. Hence, the monotonc rankng of the total traffc rate n Theorem 6 may be reversed at the ndvdual servce provder level. Fnally, as shown n Fgure 5, the optmal demand rates and servce capacty levels are monotonc n A, whch s consstent wth Propostons Concludng Remarks In ths paper, we consder a faclty provder that offers ts facltes to two servce provders, who determne ther servce capacty levels to serve two separate markets. The faclty provder can ether pool the servce provders together and let them share the facltes or allocate a dedcated faclty to the servce provders and serve them separately. We assume that the demand rate of a servce s lnearly decreasng n the total processng tme of the servce, whch s a functon of the servce capacty and faclty capacty levels. We fnd that the choce between the poolng and reservaton strateges crtcally depends on the rato of the demand loss rates of the servce provders. The faclty provder prefers the reservaton strategy to the poolng strategy f one servce provder s demand loss rate s much larger than the other s. In contrast, f the demand loss rates of the servce provders are close, the faclty provder prefers the poolng strategy. The servce provders preferences are dfferent from the faclty provder s. The domnant servce provder, whose demand loss rate s four tmes larger than the other s, prefers dedcated facltes. In contrast, the smaller servce provder, whose demand loss rate s less than the other s, prefers the poolng strategy. We also study a centralzed system, n whch the faclty provder fully controls the servce provders and hence s able to elmnate the negatve effect of the faclty competton between the servce provders. We fnd that poolng s always optmal for the faclty provder, whch suggests that faclty capacty competton s a prerequste condton for not poolng the servce provders. There are three drectons n whch ths research could be extended. Frst, we assume that the servce provders maxmze ther demand rates, as cargo volume s one of the most mportant performance measures n the martme ndustry. Another mportant performance measure for servce provders s proft. It s an nterestng research queston to desgn a jont optmal prcng polcy and a capacty management strategy for a port authorty who deals wth proft-maxmzng carrers. Second, we assume that there are two servce provders, whch s suffcent to show the tradeoff between the capacty reservaton and poolng strateges. But, n practce, a port often serves multple carrers, who may beneft from formng allances to share dedcated port facltes. Ths research queston can be studed under a cooperatve game framework. Fnally, seaports face heavy competton from local compettors. Capacty reservaton can be used as a strategc weapon to attract the busness of carrers. A game theoretcal model can be formed to study competton among seaports.