Empty container management and coordination in intermodal transport

Accepted Manuscript

Empty Container Management and Coordination in Intermodal Transport Yangyang Xie, Xiaoying Liang, Lijun Ma, Houmin Yan PII: DOI: Reference:

S0377-2217(16)30608-7 10.1016/j.ejor.2016.07.053 EOR 13879

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

15 July 2015 29 May 2016 25 July 2016

Please cite this article as: Yangyang Xie, Xiaoying Liang, Lijun Ma, Houmin Yan, Empty Container Management and Coordination in Intermodal Transport, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.07.053

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Highlights • We study empty container sharing problem in sea-rail intermodal transport. • We characterize the optimal policy for empty container delivery. • We prove the existence of unique Nash equilibrium of the stochastic inventory game.

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• We design a contract to coordinate the sea-rail intermodal transport system.

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• We show the notable performance improvement of coordination with numerical study.

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Empty Container Management and Coordination in Intermodal Transport Yangyang Xiea,b , Xiaoying Liangb , Lijun Mac,∗, Houmin Yanb a

Department of Automation, Tsinghua University, Beijing 100084, China Department of Management Sciences, College of Business, City University of Hong Kong, Hong Kong c Department of Management Science, College of Management, Shenzhen University, Shenzhen 518060, China

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b

Abstract

In this paper, we study the empty container inventory sharing and coordination problem in

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intermodal transport. We focus on dry ports in intermodal transport to raise the coordination issue in empty container management. We consider an intermodal transport system composed of one railway transport firm at a dry port and one liner firm at a seaport. First, we characterize the optimal delivery policy between the dry port and seaport in the centralized model. We investigate how the optimal policy changes with the initial inventories of

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empty containers at the dry port and seaport. Next, we design a bilateral buy-back contract to coordinate the decentralized system. We derive the Nash equilibrium of the inventory

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sharing game between the rail firm and liner firm under the decentralized model as well as the equilibrium delivery quantity with a given bilateral buy-back contract. Moreover, we

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coordinate the decentralized system by choosing appropriate contract parameters and show how the system’s profit can be distributed between the two firms under coordination.

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Keywords: Transportation; Empty container management; Contract coordination; Intermodal transport; Dry ports

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1. Introduction

Ocean shipping is a common means of freight transportation in international trade. Due

to the asymmetric nature of global trade, some transport terminals accumulate empty containers, and some others are short on container supply. As a result, the study of empty ∗

Corresponding author. Tel.: +86 755 26535181. Email addresses: [email protected] (Yangyang Xie), [email protected] (Xiaoying Liang), [email protected] (Lijun Ma), [email protected] (Houmin Yan) Preprint submitted to European Journal of Operational Research

July 28, 2016

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container repositioning has received increasing attention. We raise the coordination problems in empty container repositioning by considering dry ports in intermodal transport. As global trade volume grows, especially in developing countries, a shortage of port handling capacity has become a serious problem and led to congestion at ports. To deal with the problem, countries with high trade volumes or growing trade potential like China and India

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have begun to construct inland transport terminals – dry ports. According to the definition proposed by Leveque and Roso (2002), a dry port is “an inland intermodal terminal directly connected to seaport(s) with high capacity transport mean(s), where customers can leave/pick up their standardised units as if directly to a seaport.” Take China as an example, tens of dry ports have recently been constructed in Beijing, Shanxi, Shandong, Henan, Hebei,

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Inner Mongolia, Ningxia and Xinjiang to serve the vast inland regions of China, and more are under construction1 . China Railway Container Transport (CRCT) and China Railway Tielong Container Logistics (CRT) provide rail freight transport services between dry ports and seaports. China Ocean Shipping Company (COSCO) and China Shipping Container Lines (CSCL) provide liner services between seaports. These firms are all owners of contain-

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ers. Note that most rail firms in China own empty containers themselves. This fact can be verified by comparing company codes in the container owner list on China Railway’s official

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site2 with those codes in the registered container operator list on Bureau International des Containers et du Transport Intermodal’s official website3 . In addition, the standard and

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most commonly used sizes of inland containers in China are 20 and 40 feet4 , which are the same as those used in maritime transport. Rail and liner firms together perform intermodal

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transport. There is a motivation for the two kinds of firms to cooperate. Their cooperation helps the firms to share the resources of empty containers between them, improve the

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satisfaction level of container demands and increase the profits of both sides. Dry ports are playing an increasingly important role in global trade and have become

a hot research topic. An increasing number of works have focused on the functions and 1

http://report.hebei.com.cn/system/2013/09/09/012964753.shtml http://hyfw.95306.cn/hyinfo/action/ClcscxAction_index?cllx=JZX 3 https://www.bic-code.org/index.php?option=com_bic&task=list&Itemid=340 4 http://hyfw.95306.cn/gateway/DzswD2D/Dzsw/action/WyfhAction_initMorWyfhTbxq;DZSW_ SESSIONID=tLVXWpqcGnVFMGdM52q7ZH1T4sCJkvQSjH6wtSMhlPnnM2JRtFpP!1426081485?type=jzx 2

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implementation of dry ports from different angles, e.g., from an environmental perspective (Roso, 2007; L¨attil¨a et al., 2013), as an intermodal node (Jarˇzemskis and Vasiliauskas, 2007), within transport networks (Roso et al., 2009), in terms of the extension of port life cycles (Cullinane and Wilmsmeier, 2011) and from its development paths (Beresford et al., 2012).

the general concept and functions of dry ports.

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In this paper, we concentrate on a specific issue (empty container management) rather than

In contrast to the extensive literature that mainly discusses centralized empty container sharing and repositioning problems, we focus on a decentralized model and its coordination in empty container management. The decentralized setting is more realistic and of more value without a central planner. In this case, to implement efficient empty container management,

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firms must rely on contracts to achieve coordination. Few works have considered coordination issues in empty container repositioning. When empty containers in different ports belong to the same liner firm, the firm will conduct empty container repositioning among all of the ports as a central planner, eliminating the need for coordination with contracts. Furthermore, when empty containers belong to different liner firms, these firms become rivals to each other.

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Each firm wants its containers to be returned as soon as possible to fulfill its own demand for empty containers rather than to help its competitors. In practice, liner firms rarely lease

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empty containers from one another. We introduce the idea of rail firms at dry ports in intermodal transport to highlight the incentive for coordination.

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We assume that dry ports are connected to seaports by railway, which has a higher throughput efficiency and lower CO2 emissions (Roso, 2007). We study the empty container

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sharing and coordination problem between a liner firm at a seaport and a rail firm at a dry port in intermodal transport. The rail firm transports cargoes from the dry port to

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the seaport so that the liner firm can ship these cargoes to their destinations on the other side of the ocean. The more loaded containers the rail firm brings to the seaport, the more loaded containers are shipped out by the liner firm. There is an incentive for the liner firm and the rail firm to coordinate with each other. The relationship between liner firms at seaports and rail firms at dry ports is more like a partnership rather than a rivalry. We first show the optimal empty container repositioning and replenishment policies for the centralized system. For the decentralized model, the two firms sign a contract to determine 3

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the transfer payment for empty containers. There is an inventory sharing game between the liner firm and the rail firm. Each firm decides its capacity of empty container supply (or empty container requirement). We prove the existence of the Nash equilibrium. In addition, we design a bilateral buy-back contract to coordinate the system and distribute the system

improves the performance of the system. 2. Literature Review

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profit between the two firms. We provide numerical examples showing how the coordination

This study mainly depends on literature examining empty container repositioning, inventory sharing games and contract coordination.

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Studies of empty container repositioning have mostly formulated the problem as a linear programming (deterministic demand) or stochastic programming (stochastic demand) problem and developed algorithms to obtain the optimal or suboptimal solution. Deterministic demand models of empty container repositioning have been examined since a study by Er-

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mol’ev et al. (1976). Choong et al. (2002) present the effect of planning horizon length on empty container repositioning for intermodal transportation networks. Olivo et al. (2005)

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propose a new mathematical programming method to solve the empty container repositioning problem in a multiperiod network model. They introduce dummy nodes and arcs into the network model to offer a better interpretation of the computational results. Jula et al. (2006)

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compare two common methods of empty container reuse: street-turn (empty containers could be sent directly from local consignees to local shippers) and depot-direct (empty containers

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could be stored and interchanged at off-dock depots). If the time cost is high, then the depot-direct method would be a better choice, as it minimizes the waiting time. When the

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traveling cost is high and there is a severe traffic congestion, the street-turn method provides a better match of supply and demand. Shintani et al. (2007) study the container shipping network design problem while considering empty container repositioning. The deployments of ships and containers are considered simultaneously. For stochastic demand models, Cheung and Chen (1998) study the solution methods of the empty container allocation problem in a two-stage stochastic network model. They adopt the stochastic linearization method and stochastic hybrid approximation procedure to solve the network optimization problem. 4

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Li et al. (2004) consider the empty container repositioning problem for a single port with a long-run average criterion. They obtain the optimal policy for importing and exporting empty containers, i.e., the (U ,D) policy, in which case empty containers are imported to U when the inventory of empty containers is less than U and exported to D when the inventory is greater than D. Otherwise, nothing is done. They extend the policy to a multi-port model

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in their latter work (Li et al., 2007) and derive an -optimal policy. They obtain the convexity of the extended policy and propose a heuristic algorithm based on the convexity to gain a feasible solution for any initial system state. Song (2006) shows the optimal empty container repositioning policy in a periodic-review two-terminal system with random demands and finite capacity. He uses the Markov decision process to model the system and characterizes

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the optimal stationary policies for both expected discounted and long-run average objective functions. Lam et al. (2007) present an approximate dynamic programming approach to obtain the exact optimal solution for the empty container relocation problem in a two-ports two-voyages model. Song and Zhang (2010) model container flow as a continuous fluid and derive the optimal empty container repositioning policy for a single port with a Markovian

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demand. Yun et al. (2011) consider the optimal empty container ordering and leasing decisions for a single-depot system. They estimate the expected cost rate and obtain the near

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optimal (s, S) policy in numerical studies. Dang et al. (2013) extend Yun et al. (2011) to a two-depot system and propose four heuristic algorithms to solve the inland positioning prob-

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lem. Ng et al. (2012) study the optimal inventory transfer policy for a two-depot system with backlogging. They establish the structural properties of the value function and show that

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the optimal policy is characterized by six switching curves. Long et al. (2012) propose two heuristic algorithms to solve a two-stage stochastic empty container repositioning problem

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by applying the sample average approximation method. Zhang et al. (2013) investigate the optimal empty container repositioning policy between multi-ports with stochastic import and export of empty containers. They show that the optimal policy for a single port can be characterized by two thresholds and further develop an algorithm to obtain an approximate repositioning policy for multi-ports based on the results of the single-port case. Our model belongs to the stochastic demand setting. The previous research focuses on the empty container management in a centralized system. In contrast, we investigate the coordination 5

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issue with contracts in a decentralized system. Decentralized empty container repositioning problems can be seen as a kind of inventory sharing game. Decentralized inventory transshipment problems have been well studied. Rudi et al. (2001) show the existence and uniqueness of the Nash equilibrium in an inventory sharing game between two firms, when transshipment prices are given. They further demonstrate

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how to choose predetermined transshipment prices to achieve a joint-profit maximization of the two firms. Lee and Whang (2002) consider a two-stage model with one manufacturer and multiple resellers. In the second stage (secondary market), resellers can trade their inventories. The authors reveal how the existence of the secondary market affects supply chain welfare through the quantity and allocation effects. Soˇsic (2006) investigates the alliance

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stability of an inventory transshipment game with myopic and farsighted retailers. She uses the largest consistent set and the equilibrium process of coalition formation approaches to measure the stability. Zhao et al. (2006) study the optimal inventory transshipment policies in a decentralized dealer network while incorporating the dealers’ demand filling and requesting decisions. They show that the optimal requesting and rationing policies are of

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threshold types. Zhao and Atkins (2009) investigate an inventory transshipment game and a consumer substitution game between competing retailers. They characterize the equilibrium

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prices and safety stocks and further show the effects of transshipment price and competition on the equilibrium. Shao et al. (2011) explore the inventory transshipment problem in a

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supply chain composed of a single manufacturer and competing retailers. They show that transshipment price plays an important role in determining both the manufacturer and re-

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tailers’ profits. They reveal that the manufacturer gains a higher profit when dealing with decentralized rather than centralized retailers. These works consider transshipment prices

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as a key factor affecting the equilibrium and thus the profits of competing firms. In contrast, we design contracts to achieve coordination and allocate the system profit between firms. Moreover, whereas other works assume that inventory transshipment occurs after demand realization, we take empty container delivery lead time into consideration. Hence, delivery occurs before demand realization in our model. To improve the performance of a decentralized intermodal transport system and benefit every firm, we must establish an transfer payment scheme between firms by designing 6

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appropriate contracts. A large body of literature investigates the design and application of certain types of contracts, e.g., vendor-managed-inventory type contracts (Fry et al., 2001), wholesale price contracts (Lariviere and Porteus, 2001), push/pull/advance-purchasediscount contracts (Cachon, 2004), revenue-sharing contracts (Giannoccaro and Pontrandolfo, 2004; Cachon and Lariviere, 2005; Govindan and Popiuc, 2014), risk-sharing contracts

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with return policies (Gan et al., 2005), purchase contracts with information updates (Huang et al., 2005), contracts for differences and two-part tariffs contracts in the electricity industry (Oliveira et al., 2013), and quality-compensation contracts in the presence of quality uncertainty (Lee et al., 2013). Some works have examined the differences and similarities between different types of contracts. Cachon (2003) reviews the advantages and disadvantages of

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different types of supply chain contracts. He discusses the coordination conditions of commonly used contracts like wholesale price, buy-back and revenue-sharing contracts in detail along with the relationships between those contracts. Lu et al. (2015) study the robustness of supply chain contracts. Robustness is measured from three aspects: consistency in the channel profit distribution, decision-making sequences and compliance regimes. The authors

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establish a framework of the consistency of contracts and analyze several types of commonly used contracts within this framework. Compared with these works, we apply contract coor-

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dination theory to the empty container repositioning problem in intermodal transport. We design contracts to coordinate the intermodal transport system and allocate profit between

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firms.

The remainder of this paper is organized as follows. In Section 2, we formulate the basic

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model. Section 3 presents the main properties of the profit function and the optimal delivery and replenishment policies under the centralized setting. Section 4 investigates the

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decentralized model. We characterize the Nash equilibrium and discuss the design of coordinating contracts. We also conduct numerical experiments to demonstrate how contracts affect the intermodal transport system. Section 5 concludes the paper with a discussion of future research directions.

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3. Model Formulation We consider an intermodal transport system that consists of two firms: one rail firm that transports cargoes on land and one liner firm that ships cargoes between ports. Both firms are owners of empty containers. We assume that the rail firm owns containers at the dry port container yard (CY) and the liner firm at the seaport CY. In practice, both the rail

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firm and liner firm may own containers at both the seaport and dry port. We assume that each firm makes internal and external decisions. The internal decision is the repositioning decision on its own empty containers in different terminals, and the external decision is the container transshipment decision between firms. The internal self-optimization process has

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been well studied by previous works (e.g., Li et al., 2004, 2007). After the self-optimization by individual firms, if there exists a demand for empty containers, they try to transship the empty containers between each other. This empty container transshipment game between different firms is what we plan to investigate in this paper. In the intermodal scenario, for the rail firm, we consider only the intermodal demand at the dry port. The demand at the

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seaport (from ports to hinterland) for the rail firm is not for intermodal transport. It is a part of the rail firm’s self-optimization process. For the intermodal demand at the dry port,

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they are loaded into empty containers, transported to the seaport by the rail firm and then shipped out by the liner firm. There is no empty container accumulation at the seaport for

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the rail firm. Therefore, for the rail firm in the intermodal scenario, neither inventory nor demand of empty containers exists at the seaport. Likewise, for the liner firm, there is no

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demand at the dry port. Hence, its empty container inventory at the dry port is either used to satisfy the demand from the rail firm, or transported back to the seaport. The lead time for the empty container transshipment from the liner firm’s inventory at the dry port to the

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rail firm is very short and therefore can be assumed zero. Hence, this transshipment decision can be made when the demand realizes. On the other hand, the lead time is relatively long if the two firms want to transship empty containers between the seaport and the dry port. When making such transshipment decisions, the demands and arrivals of empty containers are considered to be uncertain. In this paper, we focus on investigating the coordination mechanism with respect to the transshipment decision with lead time and incorporate the effect of the non-lead-time transshipment decision in the uncertainty of empty container 8

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arrivals. For consignors who want to export products to overseas markets, they transport their cargoes to the dry port container freight station (CFS) or the seaport CFS. Cargoes that come to the seaport CFS are shipped out directly. Consignors lease empty containers from and pay the ocean freight cost to the liner firm. Cargoes that come to the dry port CFS are

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firstly transported to the seaport CY by train and then shipped out by the liner firm. This is an intermodal form of transportation. In this case, consignors lease empty containers from the rail firm, and pay the railway freight cost to the rail firm and the ocean freight cost to the liner firm. In both cases, the consignor has the right to use the containers during the whole transport duration. After the transport, either the lessor (liner firm or rail firm) or the lessee

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(consignor) is responsible for getting the empty containers back, depending on the terms of the lease contract. For loaded containers imported from overseas, they are unloaded at either the seaport or the dry port. After unloading, the cargoes are received by consignees. Meanwhile, the unloaded containers are added to the empty container inventory at the corresponding terminal if they are owned by the liner firm or the rail firm. Otherwise, the

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empty containers are going to be transported back to their overseas owners. As there exists a lead time for empty container transport, the arrivals and demands of empty containers are

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usually uncertain. Similar to Li et al. (2004, 2007) and Zhang et al. (2013), we assume that the arrivals and demands of empty containers are random variables. The random demands

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of empty containers are the aggregate demands during the lead time. We illustrate the schematic diagram of the model in Figure 1.

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Let Xd denote the random demand of empty containers at the dry port CFS and Xs denote that at the seaport. Let Ys and Yd denote the random arrival of empty containers at

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the seaport CFS and dry port CFS, respectively. We place no restrictions on the dependencies of the random variables. Let rl denote the revenue from leasing unit container. For each

loaded container shipped out, the liner firm gets a freight rate of rs . The rail firm charges rd per loaded container transported from the dry port to the seaport. The holding cost per empty container at the seaport CY and dry port CY are hs and hd , respectively. The goodwill

penalty (or lost sales cost) per container is gs for the liner firm and gd for the rail firm. Let ns denote the inventory of empty containers that the liner firm owns at the seaport CY and 9

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(a) Export

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(b) Import

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Figure 1: Model Description

nd denote the inventory that the rail firm owns at the dry port CY. In the centralized model,

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the decision variable is the quantity of empty containers that are transported between the seaport and dry port. Let q denote this quantity. We define that a positive q value means

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that empty containers are transported from the seaport to the dry port and a negative q value indicates the opposite. The transportation cost per container is ct . Compared with the centralized model, the decentralized model has two extra steps before

making the empty container delivery decision. In the decentralized model, we assume that two independent operators are in the system: the liner firm and rail firm. First, they determine the contract terms between them through rounds of negotiation. Then, they each make their empty container transshipment capacity decisions to maximize their own profits. 10

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Centralized Model Timeline

Contract Determination

Transshipment Capacity Decisions (Game)

Decentralized Model Timeline

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Figure 2: Timelines

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Container Delivery

There is an inventory transshipment game between the two firms. The actual delivery quantity depends on the equilibrium of the transshipment game. We explain the detailed process of decentralized model at the beginning of Section 5. We illustrate the timelines of the centralized and decentralized models in Figure 2. For the sake of convenient reference,

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we list the notations of the centralized model as follows.

Random Variables

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Notations

demand for empty containers at the seaport

Xd

demand for empty containers at the dry port

Ys

empty container generation at the seaport

Yd

empty container generation at the dry port

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Xs

Decision Variables

number of empty containers delivered from the seaport to the dry port

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q

(from the dry port to the seaport if negative)

State Variables ns

inventory of empty containers at the seaport

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inventory of empty containers at the dry port

Parameters rl

revenue from leasing per empty container 11

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revenue from shipping out per loaded container at the seaport

rd

revenue from delivering per loaded container from the dry port to the seaport

ct

cost of transporting per empty container between the seaport and the dry port

hs

holding cost per empty container at the seaport

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holding cost per empty container at the dry port

gs

goodwill penalty per unmet demand at the seaport

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goodwill penalty per unmet demand at the dry port

4. Centralized Model

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rs

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In this section, we study the optimal empty container delivery policy when the dry port and seaport are operated by a central planner. 4.1. Optimal Delivery Policy

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Let Qs (q, ns ) denote the satisfied demand for empty containers at the seaport:

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Qs (q, ns ) = min{Xs , ns − q + Ys }. Let Is (q, ns ) denote the leftover inventory at the seaport:

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Is (q, ns ) = (ns − q + Ys − Xs )+ ,

Ls (q, ns ) = (Xs + q − ns − Ys )+ .

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Let Ls (q, ns ) denote the unsatisfied demand at the seaport:

For the dry port, we define the satisfied demand, leftover inventory and unsatisfied demand as follows, respectively: Qd (q, nd ) = min{Xd , nd + q + Yd }, Id (q, nd ) = (nd + q + Yd − Xd )+

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and Ld (q, nd ) = (Xd − nd − q − Yd )+ . The transportation cost for the empty containers is ct |q|. Based on the preceding definitions, the expected system’s profit function Π(q, ns , nd ) can

Π(q, ns , nd ) = (rs + rl )EQs (q, ns ) + (rs + rd + rl )EQd (q, nd )

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be written as follows:

−hs EIs (q, ns ) − hd EId (q, nd ) − gs ELs (q, ns ) − gd ELd (q, nd ) − ct |q|.

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In the formula, every expectation is taken over the joint distribution of all of the random variables in the corresponding function, e.g., the joint distribution of (Xs , Ys ) for EQs (q, ns ). Note that the seaport’s ocean freight revenue consists of two terms. rs EQs (q, ns ) is the revenue from shipping cargoes that come directly to the seaport and rs EQd (q, nd ) is that from the intermodal transportation with the dry port. Let r¯s = rs + hs + gs + rl , r¯d =

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rd + hd + gd + rl denote the in-all-revenues of the seaport and dry port, respectively, and c¯ = hd − hs + ct 1q>0 − ct 1q<0 denote the in-all-cost. The in-all-revenue of the seaport (dry

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port) is the total revenue increment from satisfying unit demand of empty containers at the seaport (dry port) and the in-all-cost calculates the cost increment incurred by moving an

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empty container from the seaport to the dry port. Then, similar to that on page 9 in Cachon (2003), we can rewrite the profit function using the in-all-revenue and in-all-cost as follow,

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Π(q, ns , nd )

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= (rs + hs + gs + rl )E min{Xs − Ys , ns − q} +(rs + rd + hd + gd + rl )E min{Xd − Yd , nd + q}

+(hs − hd )q − ct |q| − hs ns − hd nd − gs E(Xs − Ys ) − gd E(Xd − Yd )

(1)

+(rs + rl )EYs + (rs + rd + rl )EYd

= r¯s Ss (q, ns ) + (rs + r¯d )Sd (q, nd ) − c¯q + C(ns , nd ), where Ss (q, ns ) = E min{Xs − Ys , ns − q} and Sd (q, nd ) = E min{Xd − Yd , nd + q} are the expected inventory usage at the seaport and the dry port, respectively, and C(ns , nd ) = 13

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−hs ns − hd nd − gs E(Xs − Ys ) − gd E(Xd − Yd ) + (rs + rl )EYs + (rs + rd + rl )EYd is a constant term with respect to the delivery quantity q. From equation (1), it is clear that Π(q, ns , nd ) is not differentiable at q = 0. Lemma 1. Π(q, ns , nd ) is strictly concave in q.

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Lemma 1 indicates that there exists an optimal delivery quantity q when the inventory level ns and nd are given. Based on Lemma 1, we derive the following theorem that characterizes the optimal delivery policy for the centralized model. Define the system’s profit function when the empty containers are being delivered from the seaport to the dry port as

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Π+ (q, ns , nd ), Π+ (q, ns , nd )

= (rs + rl )[EQs (q, ns ) + EQd (q, nd )] + (rd + rl )EQd (q, nd ) −hs EIs (q, ns ) − hd EId (q, nd ) − gs ELs (q, ns ) − gd ELd (q, nd ) − ct q

Π− (q, ns , nd )

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and the profit function when the delivery is from the dry port to the seaport as Π− (q, ns , nd ),

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= (rs + rl )[EQs (q, ns ) + EQd (q, nd )] + (rd + rl )EQd (q, nd )

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−hs EIs (q, ns ) − hd EId (q, nd ) − gs ELs (q, ns ) − gd ELd (q, nd ) + ct q. Theorem 1. Given ns and nd , the optimal delivery quantity q o (ns , nd ) is

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   ns ,       qˆ(ns , nd ),   o q (ns , nd ) = 0,      q˜(ns , nd ),      −n , d

for qˆ(ns , nd ) > ns > 0, for 0 ≤ qˆ(ns , nd ) ≤ ns , for qˆ(ns , nd ) < 0 and q˜(ns , nd ) > 0, for − nd ≤ q˜(ns , nd ) ≤ 0, for q˜(ns , nd ) < −nd < 0,

in which qˆ(ns , nd ) is the optimal quantity q that maximizes Π+ (q, ns , nd ) and q˜(ns , nd ) is the optimal quantity q that maximizes Π− (q, ns , nd ). For the sake of convenience, we call the five cases of optimal policy Cases A - E from 14

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top to bottom. Theorem 1 demonstrates the optimal delivery policy for the different initial inventory states. In other words, the optimal delivery quantity q o depends on ns , the initial inventory at the seaport, and nd , the initial inventory at the dry port. To understand the policy, we split the profit function of the system into three parts, i.e., the seaport, dry port

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and transportation cost, as follows. Define the profit function of the seaport as ps (ls ) = (rs +rl )E min{Xs , ls +Ys }+rs E min{Xd , ld +Yd }−hs E(ls +Ys −Xs )+ −gs E(Xs −ls −Ys )+ , the profit function of the dry port as

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pd (ld ) = (rd + rl )E min{Xd , ld + Yd } − hd E(ld + Yd − Xd )+ − gd E(Xd − ld − Yd )+ , and the transportation cost as

−ct |q|,

where ls = ns − q is the inventory level at the seaport after delivery and ld = nd + q is that

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at the dry port. The profit function of the system can be rewritten as

d p (l ) dls s s

denote the marginal profit of the seaport and md (ld ) =

d p (l ) dld d d

PT

Let ms (ls ) =

ED

Π(q, ns , nd ) = ps (ns − q) + pd (nd + q) − ct |q|.

denote that of the dry port. The optimal delivery policy depends on the difference in the

CE

two marginal profits. It is optimal to move empty containers from the side with a lower marginal profit to the higher side, so that the entire system gains a higher profit. When moving one empty container from the seaport to the dry port with inventory levels ls at

AC

the seaport and ld at the dry port, the profit of the system increases by md (ld ) − ms (ls ). If ms (ns ) > md (nd ) + ct , then moving empty containers to the seaport is profitable until the inventory levels satisfy ms (ls ) = md (ld ) + ct or the dry port runs out of stock. The optimal delivery quantity q o = − min{nd , |˜ q |}. If ms (ns ) < md (nd ) − ct , then moving empty

containers from the seaport to the dry port increases the profit of the system. Otherwise, if |ms (ns ) − md (nd )| ≤ ct , then the profit from the inventory reallocation is less than the 15

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transportation cost. In such a case, keeping the status quo is optimal. Corollary 1. (i) 0 ≤

∂q o ∂ns

≤ 1, (ii) −1 ≤

∂q o ∂nd

≤ 0 and (iii) for q o 6= 0,

∂q o ∂ns

−

∂q o ∂nd

= 1.

Corollary 1 characterizes how the optimal delivery quantity changes with the initial inventory state. Recall that we define the positive delivery direction as that from the seaport

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to the dry port. It is clear that the optimal delivery quantity is increasing in ns and decreasing in nd , as shown in Corollary 1. In addition, properties (i) and (ii) of the corollary show that the change rate of the optimal delivery quantity is no more than that of the empty container inventory at the seaport or dry port CFS. Property (iii) indicates that if the optimal quantity is nonzero, then one more empty container at the seaport and one less at the dry port leads

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to an increment of the optimal delivery quantity by one. In other words, the inventory levels after delivery are the same for those initial inventory states with the same total inventory (summation of inventories at the seaport and dry port) and are within the same case of optimal delivery policy. Based on Theorem 1 and the corollary, we illustrate the optimal delivery policy over ns and nd in Figure 3. Figures 3(a) and 3(b) are two possible alternatives

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of the optimal policy over ns and nd , depending on the parameters of the model (i.e., rs , rd ,

ED

ct , etc.). Inventory at Dry Port nd

PT

D: q o = q˜

Inventory at Dry Port nd

CE

C: q o = 0

D: q o = q˜

C: q o = 0

B: q o = qˆ E: q o = −nd

AC

A: q o = ns

B: q o = qˆ

Inventory at Seaport ns

Inventory at Seaport ns

(a)

(b) Figure 3: Optimal Policy over ns and nd

4.2. Replenishment Policy We consider the empty container replenishment policy in this subsection. We assume that the central planner can replenish empty containers from both of the two terminals. When 16

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making the replenishment decision, the delivery decision is also taken into consideration. We define the system’s profit under optimal delivery policy as Πo (ns , nd ): Πo (ns , nd ) = Π(q o (ns , nd ), ns , nd ).

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Let qsr and qdr denote the replenishment quantities at the seaport and dry port, respectively. Let qr = qsr + qdr denote the total replenishment quantity. The unit empty container purchasing costs are cs and cd at the seaport and dry port, respectively. Let n0s and n0d represent the inventory level before replenishment at the seaport and dry port, respectively. Let n0 = n0s + n0d denote the total inventory level before replenishment.

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Lemma 2. Πo (ns , nd ) is jointly concave in ns and nd .

Lemma 2 shows that Πo (ns , nd ) is concave in ns . Hence, the seaport’s replenishment policy is of a base-stock type given nd . Likewise, Πo (ns , nd ) is concave in nd , indicating that the dry port’s replenishment policy is of a base-stock type given ns . However, a combination

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of two base-stock policies is not necessarily a base-stock type policy. The optimal total replenishment amount qro for multi-terminal replenishment is state-dependent. The following

ED

theorem characterizes the optimal multi-terminal replenishment policy in relation to the different transportation costs.

PT

Theorem 2.

(i) If ct ≤ |cs − cd |, then it is always optimal to replenish from the terminal with a lower

CE

empty container purchasing cost. The problem reduces to a single-terminal replenishment problem. The optimal policy is a base-stock policy. The optimal replenishment

AC

quantity is

  n∗ − n0 , n∗ > n0 , l l l l o qr =  0, ∗ n ≤ n0l ,

where n∗l = arg max{Πo (ns , nd ) − cl nl } and l = arg min{ci } is the index of the low nl

i∈{s,d}

purchasing-cost terminal. (ii) If ct ≥ max{rs + rd + hs + gd + rl , rs + hd + gs + rl }, then the replenishment policy is equivalent to two separate base-stock policies at the seaport and dry port, respectively. 17

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(iii) Otherwise, the optimal replenishment policy is not a base-stock type and depends on the inventory levels before replenishment, i.e., nos and nod . Property (i) of Theorem 2 reveals that the optimal total replenishment policy is of a base-stock type if the transportation cost is less than the difference of the purchasing costs i∈{s,d}

purchasing-cost terminal. Note that

n∗l

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between the seaport and dry port. Let h = arg max{ci } denote the index of the highis a decreasing function of nh . Property (ii) shows

that if the transport cost is high enough, then it is non-optimal to transport any empty containers. In this case, the dry port and seaport could be seen as two separable terminals. For each terminal, the optimal replenishment policy is a base-stock policy. Property (iii)

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gives a description of the general case, where the optimal policy is state-dependent and not of a base-stock type. 5. Decentralized Model

In this section, we discuss the coordination between the liner firm and the rail firm in

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a decentralized system. The liner firm and the rail firm sign a contract, with the aim to coordinate the system and improve the profit of both sides. In practice, the contract is not

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offered directly by one of the firms. Rather, it is adopted after multiple rounds of offers and counter offers of the two firms. When the two firms reach an agreement after the

PT

negotiation process, each firm decides its own empty container supply capacity or empty container requirement. If one firm is willing to provide empty containers and the other has

CE

a requirement, then the deal is done and empty containers are moved from one CY to the other. Otherwise, if both firms want to provide containers (or both require containers),

AC

then no deal can be reached and no empty containers are moved. A simple wholesale price contract (the supplier charges a fixed wholesale price per empty container provided) can coordinate the system, but the profit increment allocation between the firms is fixed. Hence, we want to design a contract that not only coordinates the system but also achieves profit allocation within a certain range.

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5.1. Bilateral Buy-Back Contract As the empty container transport is bilateral, the empty container lease price in the empty container sharing contract should be a bilateral price. That is, whoever leases the container pays the owner of the containers at a predetermined contract price. We use buyback, a measure of risk pooling, to strengthen the cooperative relationship between the two

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firms. We propose a bilateral buy-back contract to coordinate the system and allocate the profit increment between the two firms. Here, we assume that the buy-back does not really refer to moving the empty containers back to their owner. Rather, just the owner pays the leasee a buy-back price per leftover empty container from the owner, as a compensation for the holding cost. Note that the compensation is only paid for those leftover empty containers

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that are delivered from the other firm rather than for all of the leftovers. Hence, each firm chooses to use its own empty container inventory and the arrived empty containers prior to the containers from the other firm. The leftover empty containers stay in the leasee’s CY. This assumption is similar to the commonly used assumption in vendor-managed inventory

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models, i.e., suppliers are responsible for holding costs. As the holding costs at the seaport and dry port are different, it is reasonable to assume that the compensations (buy-back

ED

prices) charged by the two terminals are different. The transfer payment from the dry port to the seaport is defined as

PT

T (q, ns , nd , w, bs , bd )

CE

= wq + − bd E min{(nd + q + Yd − Xd )+ , q + }

(2)

−w(−q)+ + bs E{(ns − q + Ys − Xs )+ , (−q)+ },

where w is the wholesale price per empty container, bs is the buy-back price paid to the

AC

seaport (liner firm) and bd is that paid to the dry port (rail firm). We assume that the provider of empty containers pays the transportation costs, as the buyer has already been charged for a wholesale price of w. If the empty containers are transported from the seaport to the dry port, then the leasing cost wq is paid by the dry port to the seaport. For every leftover empty container, the seaport compensates the dry port for the holding cost by a buy-back price of bd . A positive transfer payment T indicates that the payment is made from the dry port to the seaport and vice versa. Such a definition is consistent with the 19

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definition that a positive q indicates the empty containers are transported from the seaport to the dry port. We write the profit function of the liner firm as follows: πs (q, ns , nd , w, bs , bd ) −bd E min{(nd + q + Yd − Xd )+ , q + }

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= (rs + rl )E min{Xs , ns − q + Ys } + rs E min{Xd , nd + q + Yd } −hs E(ns − q + Ys − Xs )+ + bs E{(ns − q + Ys − Xs )+ , (−q)+ } −gs E(Xs + q − ns − Ys )+ + (w − ct )q + − w(−q)+ ,

We write the profit function of the rail firm as follows:

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πd (q, ns , nd , w, bs , bd )

= (rd + rl )E min{Xd , nd + q + Yd } − bs E min{(ns − q + Ys − Xs )+ , (−q)+ } −hd E(nd + q + Yd − Xd )+ + bd E min{(nd + q + Yd − Xd )+ , q + } −gd E(Xd − nd − q + Yd )+ + (w − ct )(−q)+ − wq + .

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Given ns , nd and the contract parameters w, bs , bd , there is an inventory transshipment game between the liner firm and the rail firm. Each firm decides how many empty containers it

ED

requires or wants to provide. Let qs and qd denote the empty container transshipment capacity decisions of the liner firm and the rail firm, respectively. A positive qs value indicates

PT

that the liner firm wants to provide empty containers and a negative qs value indicates that the liner firm requires empty containers. For qd , it is the opposite. A positive qd value refers

CE

to the demand for empty containers and a negative qd value refers to the supply capacity. According to the preceding definitions, the quantity of empty containers transported from the seaport to the dry port is q = min{qs+ , qd+ } − min{qs− , qd− }. For example, if qs and qd

AC

are both positive, then the liner firm wants to provide empty containers and the rail firm is in need of empty containers. In this case, a deal is made between the two firms, and q = min{qs , qd } empty containers will be moved from the seaport to the dry port.

Condition 1. 0 ≤ bs ≤ r¯s and −rs ≤ bd ≤ r¯d . Condition 1 is used to ensure the concavities of the seaport’s and the dry port’s profit functions, which we use to establish the existence of the Nash equilibrium. 20

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Lemma 3. Under Condition 1, πs (q, ns , nd , w, bs , bd ) and πd (q, ns , nd , w, bs , bd ) are both strictly concave in q. Lemma 3 reveals that the liner firm and the rail firm each have a unique optimal empty container capacity, denoted by qso and qdo , respectively. From the concavity of the two firms’

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profit functions, we derive the following lemma. Lemma 4. Under Condition 1, (i) there exists a unique weakly dominant strategy for the liner firm, i.e., qs = qso , and (ii) there exists a unique weakly dominant strategy for the rail firm, i.e., qd = qdo .

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Lemma 4 shows that no matter how the other player changes its strategy, each firm has a strategy that earns a payoff at least as high as any other strategy (strictly higher for some strategy profiles). We assume that the admissible decision rule for a player is that one does not use any strategy that is weakly dominated by another. The system operates under the Nash equilibrium characterized by the following theorem.

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Theorem 3. Under Condition 1, the unique Nash equilibrium by iterated elimination of weakly dominated strategies is qs = qso and qd = qdo . The equilibrium is Pareto optimal. The

ED

corresponding delivery quantity is q e = min{(qso )+ , (qdo )+ } − min{(qso )− , (qdo )− }. Theorem 3 demonstrates the existence and uniqueness of the Nash equilibrium by as-

PT

suming that the players do not use any weakly dominated strategy. The equilibrium is also Pareto optimal, which indicates that no one can get a strictly better payoff without hurting

CE

the other’s profit.

AC

5.2. Coordination with Contract In this subsection, we assume that the system operates under the Nash equilibrium in

Theorem 3. We discuss how to choose the contract parameters of a bilateral buy-back contract to coordinate the system under voluntary compliance. That is, the system gains its maximum profit when each firm makes its decision on the basis of optimizing its own profit. First, we define the profit increments of the seaport and the dry port. Compared with the case in which no empty container is transported, the profit increment earned by the liner 21

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firm is ∆πs (ns , nd , w, bs , bd ) = πs (q o , ns , nd , w, bs , bd ) − πs (0, ns , nd , w, bs , bd ),

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that of the rail firm is ∆πd (ns , nd , w, bs , bd ) = πd (q o , ns , nd , w, bs , bd ) − πd (0, ns , nd , w, bs , bd ) and the system profit increment is

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∆Π(ns , nd ) = Π(q o , ns , nd ) − Π(0, ns , nd ).

It is clear that ∆Π(ns , nd ) = ∆πs (ns , nd , w, bs , bd ) + ∆πd (ns , nd , w, bs , bd ). Theorem 4. In each case, if the contract parameters w, bs and bd satisfy the following conditions and Condition 1, then the system is coordinated.

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Case A: r¯s F¯s (0) − rs F¯d (nd + ns ) − hs + ct ≤ w − bd Fd (nd + ns ) ≤ r¯d F¯d (nd + ns ) − hd ,

s

s

s

d

d

s

s

s

s

d

d

d

d

t

PT

s

ED

Case B: r¯s F¯s (ns − qˆ) − rs F¯d (nd + qˆ) − hs + ct = w − bd Fd (nd + qˆ) = r¯d F¯d (nd + qˆ) − hd ,   r¯ F¯ (n ) − h ≤ w − b F (n ) ≤ r¯ F¯ (n ) − r F¯ (n ) − h + c , d d d d d d d s s s s d d s t Case C:  r¯ F¯ (n ) − r F¯ (n ) − h ≤ w − b F (n ) ≤ r¯ F¯ (n ) − h + c ,

Case D: r¯s F¯s (ns − q˜) − rs F¯d (nd + q˜) − hs = w − bs Fs (ns − q˜) = r¯d F¯d (nd + q˜) − hd + ct ,

CE

Case E: r¯d F¯d (0) − hd + ct ≤ w − bs Fs (ns + nd ) ≤ r¯s F¯s (ns + nd ) − rs F¯d (0) − hs ,

AC

where Fs (·) and Fd (·) are the cumulative distribution functions of Xs − Ys and Xd − Yd

respectively, and F¯s (·) = 1−Fs (·) and F¯d (·) = 1−Fd (·) are the corresponding complementary cumulative distribution functions. In the theorem, we demonstrate the constraints on the wholesales price and the buyback prices that coordinate the system in different cases of optimal policy. Note that in Cases B and D, the latter equations are automatically satisfied by the definitions of qˆ and q˜, respectively. Hence, there is only one equation constraint each for Cases B and D. The 22

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ranges of the profit increment of both the liner firm and the rail firm under all five of the optimal policy cases are listed in Table 2. Table 2: Range of Profit Increment

A B C D E

Liner firm (∆πs ) Lower Bound Upper Bound r¯s ∆Is (q o ) + rs |∆Id (q o )| ∆Π r¯s ∆Is (q o ) + rs |∆Id (q o )| ∆Π 0 0 o rs |∆Id (q )| ∆Π − r¯d ∆Id (q o ) o rs |∆Id (q )| ∆Π − r¯d ∆Id (q o )

Rail firm (∆πd ) Lower Bound Upper Bound 0 ∆Π − r¯s ∆Is (q o ) − rs |∆Id (q o )| 0 ∆Π − r¯s ∆Is (q o ) − rs |∆Id (q o )| 0 0 o r¯d ∆Id (q ) ∆Π − rs |∆Id (q o )| o r¯d ∆Id (q ) ∆Π − rs |∆Id (q o )|

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Case

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where ∆Is (q) = E[Is (0, ns ) − Is (q, ns )] − qFs (ns − q) is the expectation of the efficient leftover inventory loss between the scenarios with and without container transport at the seaport and ∆Id (q) = E[Id (0, nd ) − Id (q, nd )] + qFd (nd + q) is that at the dry port. In Cases A and B, the seaport is the supplier of empty containers. Hence, the upper bound of the profit increment of the liner firm is the entire profit increment of the system. If the liner firm earns more than ∆Π, which is the entire profit increment, then it is clear

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that the dry port’s profit falls below the level before coordination. In this case, the rail firm

ED

would reject the contract offer and run the business on its own. The lower bound of the profit increment of the liner firm consists of two parts. One is from the empty container leasing and the other is from shipping out the returning containers via intermodal transportation.

PT

In this table, ∆Is (q o ) is the expected efficient leftover inventory loss of the seaport. If the demand at the seaport is realized at a lower level than ns − q o , then there is at least q o

CE

inventory left after the demand is fulfilled. The delivery decision does not cause goodwill penalty. In this case, we call the delivered empty containers inefficient inventory for the

AC

seaport. The liner firm does not ask for a profit for the loss of this inefficient inventory in the lower bound case of profit allocation. The expectation of such inefficient inventory loss is q o Fs (ns − q o ). However, if the demand is realized at a level higher than ns − q o , then

the delivery incurs a goodwill penalty for the liner firm. The liner firm requires at least a profit of r¯s for unit leftover inventory loss. In addition, the rail firm gains |∆Id (q o )| efficient inventory from the delivery. These empty containers are loaded at the dry port and again transported to the seaport. Shipping such returning containers brings the liner firm an profit 23

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increment of rs per container. In Case C, no delivery is optimal for the system. Hence, there is no profit allocation problem between the liner firm and the rail firm. In Cases D and E, the empty container flow is from the dry port to the seaport. The rail firm is the supplier of empty containers. It earns an profit increment of r¯d ∆Id (q) at least and ∆Π − rs |∆Id (q o )| at most, where ∆Id (q) is the expected efficient leftover inventory loss of the dry port. We can

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see that the liner firm takes more advantage of the coordination in the intermodal transport. It earns profit from not only the seaport’s demand but also the intermodal transport with the rail firm. 6. Numerical Examples

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In this section, we give numerical examples to show how the coordination improves the performance of the system and the profits of each firm. The ocean freight rs is higher than the rail freight rd . Dry ports commonly have more usable land than seaports. Thus, the holding cost at the dry port hd is lower than that of the seaport hs . Xs , Ys , Xd and Yd

M

are normally distributed. Since the dry port is constructed to solve the congestions in the seaport, in most of the real cases, seaports are more congested than dry ports. Hence, in

ED

this example, we choose parameters under which the seaport faces higher levels of demand and arrival of empty containers than the dry port. The parameters are listed as follows.

PT

Xs ∼ N ormal(2500, 3500), Ys ∼ N ormal(500, 500),

CE

Xd ∼ N ormal(800, 1000), Yd ∼ N ormal(300, 300).

rs = 3000, hs = 200, gs = 200, rd = 1000, hd = 50, gd = 200, rl = 50, ct = 100.

AC

We determine how the system profit increment percentage

∆Π(ns ,nd ) Π(ns ,nd ,0)

changes with the seaport

and the dry port initial inventories of empty containers in Figures 4(a) and 4(b), respectively. The bounds of the seaport profit allocation percentage

∆πs (ns ,nd ,w,bs ,bd ) ∆Π(ns ,nd )

under coordination

are illustrated as well. Figure 4(a) shows the results based on the seaport initial inventory ns changing from 0 to 4000 and the fixed dry port inventory nd = 1000. As ns increases, the optimal delivery policy changes from Case D to C and then B. Under Case C, keeping the status quo is 24

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100.00%

100.00%

CaseC

90.00%

90.00%

80.00%

80.00%

CaseD

CaseB

60.00%

ProfitAllocation LowerBoundof Seaport(%)

50.00% 40.00%

SystemProfit Increment(%)

30.00%

30.00%

10.00%

2000

3000

CaseC

40.00%

10.00% 1000

݊௦

0

(a) Given nd = 1000

SystemProfit Increment(%)

CaseB

0.00%

4000

ProfitAllocation LowerBoundof Seaport(%)

50.00%

20.00%

0

CaseD

60.00%

20.00%

0.00%

ProfitAllocation UpperBoundof Seaport(%)

70.00%

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ProfitAllocation UpperBoundof Seaport(%)

70.00%

1000

2000

3000

݊ௗ

4000

(b) Given ns = 1000

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Figure 4: Profit Increment and Bounds of Profit Allocation

optimal. Hence, the system profit increment is 0. Under Case D, the profit increment percentage is up to 40%. Under Case B, the profit increment is negligible because we set nd = 1000 > E(Xd − Yd ) = 500 and the delivery direction is from the seaport to the dry port. The dry port has ample empty containers to fulfill its own demand. Therefore, the

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marginal profit at the dry port is low. Figure 4(b) shows the results based on the dry port initial inventory changing from 0 to 4000 and the fixed seaport inventory ns = 1000. As nd

ED

increases, the optimal delivery policy changes from Case B to C and then D. Under both Case B and D, the system profit can be improved as much as 80%. In both Figure 4(a)

PT

and 4(b), the profit allocation lower bounds of the liner firm are higher than 40%. This is because that the liner firm makes profits from both the seaport’s and dry port’s demands,

CE

while the rail firm can only profit from the dry port’s demand.

AC

7. Conclusion

In this paper, we investigate the empty container delivery problem in an intermodal

transport system composed of a liner firm and a rail firm. For the centralized model, we derive the optimal delivery policy and show how the policy changes with the initial inventory state. The optimal delivery direction and quantity depend on the difference in marginal profits between the seaport and dry port. Empty containers are moved from the side with a lower marginal profit to the higher side. We also show the optimal inventory replenishment 25

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policy is not of a base-stock type except under specific conditions. For the decentralized model, we design a bilateral buy-back contract to coordinate the system. We derive the equilibrium delivery quantity of the inventory transshipment game between the two firms when a contract is given. We reveal how to choose the appropriate contract parameters to coordinate the system and allocate the system profit between the liner firm and the rail firm.

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The profit allocation ranges under all of the optimal policy cases are demonstrated in Table 2.

Future research may consider the multi-period intermodal transport system coordination problem with contracts renewed periodically. In the multi-period model, we need to take the contract parameters’ effect on the initial inventory state of the next period into considera-

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tion when choosing the terms. We may derive the optimal delivery policy using backward induction for the centralized model. However, for the decentralized model, each firm makes its delivery capacity decision based on the total expected profit instead of the one-period profit. Therefore, the range of profit allocation changes. The problem is more challenging to solve.

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Another interesting direction is to explore the intermodal transport system with multiple seaports. In this model, we assume that the consignor leases containers from the rail firm in

ED

the intermodal transport case. These containers are transported to the overseas destinations. Commonly, the rail firm does not run business overseas. Hence, there is a need for the rail

PT

firm to get the empty containers back. If we take the overseas destination seaports into consideration, the rail firm could sign an agreement with the liner firm to share its containers

CE

in a certain extent. The liner firm could help the rail firm to get the unloaded containers back from overseas, and the rail firm might allow the liner firm to use these containers with a

AC

low price or even for free. We can similarly design coordination contracts to bring a win-win situation for the two firms. Batch transportation of containers is also an important practical issue to be addressed.

Let B denote the batch size of the rail firm. The problem changes from a continuous optimization to a discrete optimization with interval B. Since the objective function is concave in delivery quantity (Lemma 1), the discrete optimal quantity should be one of the two nearest feasible points to the continuous optimal solution. After obtaining the optimal 26

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solution of the discrete problem, we may similarly design contracts to coordinate the system and distribute the system profit at the discrete optimal point. It would also be interesting to take the container shipping capacity into consideration. Let Crs and Crd denote the empty container shipping capacity constraints from the seaport to the dry port and from the dry port to the seaport, respectively. By assuming that the

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empty containers start to be transported right after the transshipment decision is made, Crs and Crd are deterministic values. We can incorporate the constraints by replacing ns and nd in our model with min{ns , Crs } and min{nd , Crd }. As ns , nd , Crs , Crd are deterministic, all the properties and policy structure we derived in this paper preserve.

The port/dry port container-handling capacity constraint is another interesting issue to

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consider. Container-handling capacity is defined as the total number of containers that can be handled per unit of time. The liner firm must make additional decisions about the import (unloading) container quantity and export (loading) quantity after demand realization. Let qi and qo denote the import and export container quantities, respectively. qi is the number of containers added to the empty container inventory. qo is the number of loaded containers

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shipped out, which the liner firm earns profit from. Let xs and ys denote the realizations of Xs and Ys , respectively. The ys − qi cargo containers must wait to be unloaded and the

ED

xs − qo containers of cargoes wait to be loaded into empty containers and shipped out. We

CE

PT

can write the container-handling capacity constraint of the seaport as follows: q o ≤ xs , q i ≤ ys , qo + qi ≤ Hs ,

AC

where Hs is the handling capacity of the seaport. The problem becomes a two-stage decision model. The empty container delivery capacity decision is made before realization and the import/export quantity decisions are made after realization. The dry port faces a similar capacity constraint, but its capacity is commonly much larger than that of the seaport. To simplify the problem, we may first neglect the constraint for the dry port.

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Acknowledgements

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The authors are grateful to the area editor and three anonymous referees for their insightful comments, which have helped to improve the exposition of this paper significantly. The authors thank Prof. Jan Fransoo and his student Mr. Stefano Fazi for their helpful comments on the early version of this paper. This research is supported in part by GRF Grants 4187/09E, RGC Theme-based Research Scheme T32-620/11, City University of Hong Kong Start-up Grant 7200290, NSFC Grants 71271182, 71302189 and 71471118, the Humanities and Social Sciences Foundation of Ministry of Education of China (No. 14YJC630096), and the Distinguished University Young Scholar Program of Guangdong Province (No. Yq2013140). References References

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G. P. Cachon. The allocation of inventory risk in a supply chain: Push, pull, and advancepurchase discount contracts. Management Science, 50(2):222–238, 2004.

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G. P. Cachon and M. A. Lariviere. Supply chain coordination with revenue-sharing contracts: strengths and limitations. Management Science, 51(1):30–44, 2005.

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R. K. Cheung and C.-Y. Chen. A two-stage stochastic network model and solution methods for the dynamic empty container allocation problem. Transportation Science, 32(2):142– 162, 1998.

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L. L¨attil¨a, V. Henttu, and O.-P. Hilmola. Hinterland operations of sea ports do matter: dry port usage effects on transportation costs and co 2 emissions. Transportation Research Part E: Logistics and Transportation Review, 55:23–42, 2013.

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C. H. Lee, B.-D. Rhee, and T. Cheng. Quality uncertainty and quality-compensation contract for supply chain coordination. European Journal of Operational Research, 228(3):582–591, 2013.

CE

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AN US

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X. Zhao and D. Atkins. Transshipment between competing retailers. IIE Transactions, 41 (8):665–676, 2009.

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Supplementary Appendix A. Proof of Lemma 1

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Π(q, ns , nd ) = (rs + rl )(E(min{Xs , ns − q + Ys } + E min{Xd , nd + q + Yd }) +(rd + rl )E min{Xd , nd + q + Yd } −hs E(ns − q + Ys − Xs )+ − hd E(nd + q + Yd − Xd )+ −gs E(Xs + q − ns − Ys )+ − gd E(Xd − nd − q − Yd )+ − ct |q| = (rs + hs + gs + rl )E min{Xs − Ys , ns − q} +(rs + rd + hd + gd + rl )E min{Xd − Yd , nd + q} + (hs − hd )q − ct |q| −hs ns − hd nd − gs E(Xs − Ys ) − gd E(Xd − Yd ) + (rs + rl )EYs + (rd + rl )EYd

AN US

Let Zs = Xs − Ys . Denote its probability density function (pdf) fs (·) and cumulative distribution function (cdf) Fs (·). Let Zd = Xd − Yd . Denote its pdf fd (·) and cdf Fd (·). Let F¯s (·) = 1 − Fs (·) and F¯d (·) = 1 − Fd (·) denote the corresponding complementary cdfs. For q 6= 0, ∂Π (q, ns , nd ) = −(rs +hs +gs +rl )F¯s (ns −q)+(rs +rd +hd +gd +rl )F¯d (nd +q)+hs −hd −ct sgn(q). ∂q

M

Since Π(q, ns , nd ) is not differentiable at q = 0, we show the difference between its right derivative and left derivative is negative.

ED

∂Π ∂Π |q→0+ − |q→0− = −2ct . ∂q ∂q

For other differentiable points,

PT

∂ 2Π (q, ns , nd ) = −(rs + hs + gs + rl )fs (ns − q) − (rs + rd + hd + gd + rl )fd (nd + q) < 0. ∂q 2

CE

Hence, Π(q, ns , nd ) is strictly concave in q.

AC

Appendix B. Proof of Theorem 1 From Lemma 1, we know that Π(q, ns , nd ) is concave in q. Therefore, there exists a unique optimal delivery quantity q o for the entire system, given ns and nd . The delivery quantity q is constrained in [−nd , ns ]. Let qˆ and q˜ denote the optimal delivery quantities that maximize ∂Π+ (q, ns , nd ) and ∂Π− (q, ns , nd ), respectively. That is, qˆ satisfies ∂Π+ (q, ns , nd ) = −(rs +hs +gs +rl )F¯s (ns −q)+(rs +rd +hd +gd +rl )F¯d (nd +q)+hs −hd −ct = 0, ∂q

32

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and q˜ satisfies ∂Π− (q, ns , nd ) = −(rs +hs +gs +rl )F¯s (ns −q)+(rs +rd +hd +gd +rl )F¯d (nd +q)+hs −hd +ct = 0. ∂q It is clear that q˜ > qˆ always holds. Then, we can divide the optimal policy into the following five cases.

∂Π (ns , ns , nd ) > 0. ∂q

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Case A: q˜ > qˆ > ns . Based on the definition of qˆ and q˜,

Moreover, we know that Π(q, ns , nd ) is concave in q. Hence, the optimal policy is to move all the empty containers at the seaport to the dry port, q o = ns .

AN US

Case B: 0 ≤ qˆ ≤ ns . From the condition, we derive

∂Π (ˆ q , ns , nd ) = 0. ∂q The optimal delivery quantity is qˆ.

Case C: qˆ < 0 < q˜. The condition is equivalent to

M

∂Π+ (0, ns , nd ) < 0, ∂q

and

ED

∂Π− (0, ns , nd ) > 0. ∂q

PT

The two inequalities indicate that keeping status quo is optimal.

CE

Case D: −nd ≤ q˜ ≤ 0. From the condition, we derive ∂Π (˜ q , ns , nd ) = 0. ∂q

AC

The optimal delivery quantity is q˜. Case E: qˆ < q˜ < −nd . We derive ∂Π (−nd , ns , nd ) < 0 ∂q

The optimal policy is to move all the empty containers at the dry port to the seaport, q o = −nd .

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Appendix C. Proof of Corollary 1

∂Πq (ˆ q , ns , nd ) ∂ns

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(q, ns , nd ). We prove the corollary under the five cases of the Denote Πq (q, ns , nd ) = ∂Π ∂q optimal delivery policy respectively. For Case A (q o = ns ), C (q o = 0) and E (q o = −nd ), it ∂q o ∂q o is easy to show that 0 ≤ ∂n ≤ 1 and −1 ≤ ≤ 0. Next, we show under cases B (q o = qˆ) ∂n s d and D (q o = q˜), the same results could also be derived. Case B: ∂ qˆ ∂ qˆ = (rs + hs + gs + rl )fs (ns − qˆ)(1 − ∂n ) − (rs + rd + hd + gd + rl )fd (nd + qˆ) ∂n =0 s s (rs +hs +gs +rl )fs (ns −ˆ q) ∂ qˆ ⇒ = (rs +hs +gs +rl )fs (ns −ˆq)+(rs +rd +hd +gd +rl )fd (nd +ˆq) ≤ 1. ∂ns

⇒

∂Πq ∂ qˆ (ˆ q , ns , nd ) = −(rs + hs + gs )fs (ns − qˆ) ∂n − (rs + rd ∂nd d (rs +rd +hd +gd )fd (nd +ˆ q) ∂ qˆ = − (rs +hs +gs )fs (ns −ˆq)+(rs +rd +hd +gd )fd (nd +ˆq) ≥ −1. ∂nd

AN US

Case D:

∂ qˆ + hd + gd )fd (nd + qˆ)( ∂n + 1) = 0 d

∂Πq (˜ q , ns , nd ) ∂ns

∂ q˜ ∂ q˜ = (rs + hs + gs + rl )fs (ns − q˜)(1 − ∂n ) − (rs + rd + hd + gd + rl )fd (nd + q˜) ∂n =0 s s (rs +hs +gs +rl )fs (ns −˜ q) ∂ q˜ ⇒ = (rs +hs +gs +rl )fs (ns −˜q)+(rs +rd +hd +gd +rl )fd (nd +˜q) ≤ 1. ∂ns ∂Πq (˜ q , ns , nd ) ∂nd

o

M

∂ q˜ ∂ q˜ − (rs + rd + hd + gd + rl )fd (nd + q˜)( ∂n + 1) = 0 = −(rs + hs + gs + rl )fs (ns − q˜) ∂n d d (rs +rd +hd +gd +rl )fd (nd +˜ q) ∂ q˜ ⇒ = − (rs +hs +gs +rl )fs (ns −˜q)+(rs +rd +hd +gd +rl )fd (nd +˜q) ≥ −1. ∂nd

ED

∂q We conclude that 0 ≤ ∂n ≤ 1 and −1 ≤ s ∂q o ∂q o For Case A, B, D and E, ∂ns − ∂n = 1. d

∂q o ∂nd

≤ 0 hold under all cases of the optimal policy.

PT

Appendix D. Proof of Lemma 2

CE

To prove that Πo (ns , nd ) is jointly concave in ns and nd is equivalent to prove that the Hessian matrix of Πo (ns , nd ) is negative-semidefinite. The system profit function is

AC

Π(q, ns , nd ) = (rs + rl )[EQs (q, ns ) + EQd (q, nd )] + rd EQd (q, nd ) −hs EIs (q, ns ) − hd EId (q, nd ) − gs ELs (q, ns ) − gd ELd (q, nd ) − ct |q| = (rs + rl )(E min{Xs , ns − q + Ys } + E min{Xd , nd + q + Yd }) + rd E min{Xd , nd + q + Yd } −hs E(ns − q + Ys − Xs )+ − hd E(nd + q + Yd − Xd )+ −gs E(Xs + q − ns − Ys )+ − gd E(Xd − nd − q − Yd )+ − ct |q| = (rs + hs + gs + rl )E min{Xs − Ys , ns − q} + (rs + rd + hd + gd + rl )E min{Xd − Yd , nd + q} +(hs − hd )q − ct |q| − hs ns − hd nd − gs E(Xs − Ys ) − gd E(Xd − Yd ) +(rs + rl )EYs + (rs + rd + rl )EYd .

34

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The first order derivatives are ∂Πo (ns , nd ) ∂ns

∂q o ∂q o = (rs + hs + gs + rl )F¯s (ns − q o )(1 − ∂n ) + (rs + rd + hd + gd + rl )F¯d (nd + q o ) ∂n s s ∂q o o ∂q o − c sgn(q ) − h . +(hs − hd ) ∂n t s ∂ns s ∂Πo (ns , nd ) ∂nd

∂q o ) ∂nd

CR IP T

∂q o = (rs + hs + gs + rl )F¯s (ns − q o )(− ∂n ) + (rs + rd + hd + gd + rl )F¯d (nd + q o )(1 + d ∂q o o ∂q o +(hs − hd ) ∂nd − ct sgn(q ) ∂nd − hd .

The second order derivatives are ∂ 2 Πo (ns , nd ) ∂n2s

2 o ∂q o 2 ) + (rs + hs + gs + rl )F¯s (ns − q o )(− ∂∂nq2 ) ∂ns s ∂q o 2 ¯d (nd + q o ) ∂ 2 q2o −(rs + rd + hd + gd + rl )fd (nd + q o )( ∂n ) + (r + r + h + g + r ) F s d d d l ∂ns s ∂ 2 qo o ∂q o 2 o ∂ 2 qo +(hs − hd ) ∂n2 − 2ct δ(q )( ∂ns ) − ct sgn(q ) ∂n2 . s s

∂ 2 Πo (ns , nd ) ∂ns ∂nd

2

AN US

= −(rs + hs + gs + rl )fs (ns − q o )(1 −

o

= ∂n∂ dΠ∂ns (ns , nd ) 2 qo ∂q o ∂q o ) = (rs + hs + gs + rl )fs (ns − q o )(1 − ∂n − (rs + hs + gs + rl )F¯s (ns − q o ) ∂n∂d ∂n s ∂nd s ∂q o ∂q o ¯d (nd + q o ) ∂ 2 qo −(rs + rd + hd + gd + rl )fd (nd + q o ) ∂n (1 + ) + (r + r + h + g + r ) F s d d d l ∂nd ∂nd ∂ns s 2 qo o ∂ 2 qo o ∂q o ∂q o +(hs − hd ) ∂n∂d ∂n − c sgn(q ) − 2c δ(q ) . t t ∂ns ∂nd ∂nd ∂ns s

M

∂ 2 Πo (ns , nd ) ∂n2d

2 o ∂q o 2 ) + (rs + hs + gs + rl )F¯s (ns − q o )(− ∂∂nq2 ) = −(rs + hs + gs + rl )fs (ns − q o )( ∂n d d o 2 o −(rs + rd + hd + gd + rl )fd (nd + q o )(1 + ∂q )2 + (rs + rd + hd + gd + rl )F¯d (nd + q o ) ∂ q2

∂nd 2 o ct sgn(q o ) ∂∂nq2 . d

ED

+(hs −

2 o hd ) ∂∂nq2 d

− 2ct δ(q

∂q o 2 ) )( ∂n d

−

PT

Case A: q o = ns .

o

CE

∂Πo (ns , nd ) = (rs + rd + hd + gd + rl )F¯d (nd + ns ) − hd − ct . ∂ns

AC

∂Πo (ns , nd ) = (rs + rd + hd + gd + rl )F¯d (nd + ns ) − hd . ∂nd

∂ 2 Πo (ns , nd ) = −(rs + rd + hd + gd + rl )fd (nd + ns ) < 0 ∂n2s

∂ 2 Πo ∂ 2 Πo (ns , nd ) = (ns , nd ) = −(rs + rd + hd + gd + rl )fd (nd + ns ) ∂ns ∂nd ∂nd ∂ns ∂ 2 Πo (ns , nd ) = −(rs + rd + hd + gd + rl )fd (nd + ns ) ∂n2d

35

∂nd

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In this case,



∂ 2 Πo ∂n2s ∂ 2 Πo ∂nd ∂ns

∂ 2 Πo ∂ns ∂nd ∂ 2 Πo ∂n2d

= 0,

the Hessian matrix of Πo (ns , nd ) is negative-semidefinite. Case B: q o = qˆ.

∂ qˆ ∂ qˆ = (rs + hs + gs + rl )F¯s (ns − qˆ)(1 − ∂n ) + (rs + rd + hd + gd + rl )F¯d (nd + qˆ) ∂n s s ∂ qˆ − h +(hs − hd − ct ) ∂n s s = (rs + hs + gs + rl )F¯s (ns − qˆ) − hs = (rs + rd + hd + gd + rl )F¯d (nd + qˆ) − hd − ct

CR IP T

∂Πo (ns , nd ) ∂ns

∂Πo (ns , nd ) ∂nd

= (rs + rd + hd + gd + rl )F¯d (nd + qˆ) − hd = (rs + hs + gs + rl )F¯s (ns − qˆ) − hs + ct

∂ 2 Πo (ns , nd ) ∂ns ∂nd

2

AN US

∂ 2 Πo ∂ qˆ )<0 (ns , nd ) = −(rs + hs + gs + rl )fs (ns − qˆ)(1 − 2 ∂ns ∂ns o

= ∂n∂ dΠ∂ns (ns , nd ) ∂ qˆ ∂ qˆ = (rs + hs + gs + rl )fs (ns − qˆ) ∂n = −(rs + rd + hd + gd + rl )fd (nd + qˆ) ∂n s d

M

∂ qˆ ∂ 2 Πo (ns , nd ) = −(rs + rd + hd + gd + rl )fd (nd + qˆ)(1 + ) 2 ∂nd ∂nd As shown in Corollary 1,

ED

∂ qˆ (rs + hs + gs + rl )fs (ns − qˆ) ≤ 1, = ∂ns (rs + hs + gs + rl )fs (ns − qˆ) + (rs + rd + hd + gd + rl )fd (nd + qˆ) and

CE

In this case,

PT

∂ qˆ (rs + rd + hd + gd + rl )fd (nd + qˆ) ≥ −1. =− ∂nd (rs + hs + gs + rl )fs (ns − qˆ) + (rs + rd + hd + gd + rl )fd (nd + qˆ)

∂ 2 Πo ∂n2s ∂ 2 Πo ∂nd ∂ns

∂ 2 Πo ∂ns ∂nd ∂ 2 Πo ∂n2d

= 0,

AC

the Hessian matrix of Πo (ns , nd ) is negative-semidefinite.

Case C: q o = 0.

∂Πo (ns , nd ) = (rs + hs + gs + rl )F¯s (ns ) − hs . ∂ns ∂Πo (ns , nd ) = (rs + rd + hd + gd + rl )F¯d (nd ) − hd . ∂nd ∂ 2 Πo (ns , nd ) = −(rs + hs + gs + rl )fs (ns ) < 0 ∂n2s 36

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∂ 2 Πo ∂ 2 Πo (ns , nd ) (ns , nd ) = =0 ∂ns ∂nd ∂nd ∂ns

In this case,



∂ 2 Πo ∂n2s ∂ 2 Πo ∂nd ∂ns

∂ 2 Πo ∂ns ∂nd ∂ 2 Πo ∂n2d

> 0,

the Hessian matrix of Πo (ns , nd ) is negative-definite. Case D: q o = q˜.

∂ q˜ ∂ q˜ = (rs + hs + gs + rl )F¯s (ns − q˜)(1 − ∂n ) + (rs + rd + hd + gd + rl )F¯d (nd + q˜) ∂n s s ∂ q˜ − hs +(hs − hd + ct ) ∂n s = (rs + hs + gs + rl )F¯s (ns − q˜) − hs = (rs + rd + hd + gd + rl )F¯d (nd + q˜) − hd + ct ∂Πo (ns , nd ) ∂nd

AN US

∂Πo (ns , nd ) ∂ns

CR IP T

∂ 2 Πo (ns , nd ) = −(rs + rd + hd + gd + rl )fd (nd ) ∂n2d

= (rs + rd + hd + gd + rl )F¯d (nd + q˜) − hd = (rs + hs + gs + rl )F¯s (ns − q˜) − hs − ct

∂ 2 Πo ∂ q˜ )<0 (ns , nd ) = −(rs + hs + gs + rl )fs (ns − q˜)(1 − 2 ∂ns ∂ns ∂ 2 Πo (ns , nd ) ∂ns ∂nd

2

o

M

= ∂n∂ dΠ∂ns (ns , nd ) ∂ q˜ ∂ q˜ = (rs + hs + gs + rl )fs (ns − q˜) ∂n = −(rs + rd + hd + gd + rl )fd (nd + q˜) ∂n s d

ED

∂ 2 Πo ∂ q˜ ) (ns , nd ) = −(rs + rd + hd + gd + rl )fd (nd + q˜)(1 + 2 ∂nd ∂nd

PT

As shown in Corollary 1,

and

CE

∂ q˜ (rs + hs + gs + rl )fs (ns − q˜) = ≤ 1, ∂ns (rs + hs + gs + rl )fs (ns − q˜) + (rs + rd + hd + gd + rl )fd (nd + q˜)

AC

∂ q˜ (rs + rd + hd + gd + rl )fd (nd + q˜) =− ≥ −1. ∂nd (rs + hs + gs + rl )fs (ns − q˜) + (rs + rd + hd + gd + rl )fd (nd + q˜)

In this case,



∂ 2 Πo ∂n2s ∂ 2 Πo ∂nd ∂ns

∂ 2 Πo ∂ns ∂nd ∂ 2 Πo ∂n2d

= 0,

the Hessian matrix of Πo (ns , nd ) is negative-semidefinite. Case E: q o = −nd .

∂Πo (ns , nd ) = (rs + hs + gs + rl )F¯s (ns + nd ) − hs . ∂ns 37

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∂Πo (ns , nd ) = (rs + hs + gs + rl )F¯s (ns + nd ) − hs − ct . ∂nd ∂ 2 Πo (ns , nd ) = −(rs + hs + gs + rl )fs (ns + nd ) < 0 ∂n2s ∂ 2 Πo ∂ 2 Πo (ns , nd ) = (ns , nd ) = −(rs + hs + gs + rl )fs (ns + nd ) ∂ns ∂nd ∂nd ∂ns

In this case,



∂ 2 Πo ∂n2s ∂ 2 Πo ∂nd ∂ns

∂ 2 Πo ∂ns ∂nd ∂ 2 Πo ∂n2d

= 0,

CR IP T

∂ 2 Πo (ns , nd ) = −(rs + hs + gs + rl )fs (ns + nd ) ∂n2d

Appendix E. Proof of Theorem 2

AN US

the Hessian matrix of Πo (ns , nd ) is negative-semidefinite. Based on above arguments, we have shown that the Hessian matrix of Πo (ns , nd ) is pieceo o (ns , nd ) and ∂Π (ns , nd ) wise negative-semidefinite. In addition, from the expressions of ∂Π ∂ns ∂nd ∂Πo ∂Πo in the five cases, it is clear that ∂ns (ns , nd ) and ∂nd (ns , nd ) are continuous. Therefore, we conclude that the Hessian matrix of Πo (ns , nd ) is negative-semidefinite.

PT

ED

M

Appendix E.1. Property (i) If the transport cost is negligible, replenishing containers from the terminal with lower purchasing cost and then deliver to the other terminal is always more cost-efficient than replenishing from the high purchasing-cost terminal. The problem reduces to a single-terminal replenishment problem from the low purchasing-cost terminal. From Lemma 2, we know that Πo (ns , nd ) is concave in both ns and nd . Hence, Πo (ns , nd ) is concave in nl , no matter l = s or d. The optimal policy for the single-terminal replenishment is a base-stock policy.

AC

CE

Appendix E.2. Property (ii) From the proof of Theorem 1, under assumption ct ≥ max{rs + rd + hs + gd + rl , rs + hd + gs + rl }, the optimal delivery policy is keeping status quo. Hence, the replenishment problem at the two terminals can be solved separately. Appendix E.3. Property (iii) We give a simple counter example to show the optimal policy is not of a base-stock type. Assume cs > ct . We use a proof by contradiction. Assume the optimal policy is of a base-stock type and the base-stock levels for the seaport and the dry port are n∗s and n∗d , respectively. Then, for initial inventory state, n0s = 0 and n0d = n∗s + n∗d , the optimal policy should be to replenish n∗s at the seaport and not to replenish at the dry port. In this case, the system profit is Πo (n∗s , n∗s + n∗d ) − cs n∗s . However, if the central planner chooses to transport n∗s from the dry port to the seaport and does not replenish containers, the system profit is Πo (n∗s , n∗d ) − ct n∗s . Since the optimal policy is of a base-stock type, we have 38

ACCEPTED MANUSCRIPT

Πo (n∗s , n∗d ) ≥ Πo (n∗s , n∗s + n∗d ). With cs > ct , we get Πo (n∗s , n∗d ) − ct n∗s > Πo (n∗s , n∗s + n∗d ) − cs n∗s . Here comes the contradiction. Hence, the optimal policy is not of a base-stock type. Appendix F. Proof of Lemma 3 The profit function of the liner firm is

AN US

∂ π (q, ns , nd , w, bs , bd ) ∂q s

CR IP T

πs (q, ns , nd , w, bs , bd ) = (rs + rl )E min{Xs , ns − q + Ys } + rs E min{Xd , nd + q + Yd } − bd E min{(nd + q + Yd − Xd )+ , q + } −hs E(ns − q + Ys − Xs )+ + bs E{(ns − q + Ys − Xs )+ , (−q)+ } −gs E(Xs + q − ns − Ys )+ + (w − ct )q + − w(−q)+ = (rs + hs + gs + rl − bs 1q<0 )E min{Xs − Ys , ns − q} + (rs + bd 1q>0 )E min{Xd − Yd , nd + q} −(−hs + bs 1q<0 + bd 1q>0 )q +(w − ct )q + − w(−q)+ + bd 1q>0 E(nd + Yd − Xd )+ − bs 1q<0 E(ns + Ys − Xs )+ −(hs − bs 1q<0 )ns − bd 1q>0 nd − gs E(Xs − Ys ) + rs (EYs + EYd ) = −(rs + hs + gs + rl − bs 1q<0 )F¯s (ns − q) + (rs + bd 1q>0 )F¯d (nd + q) −(−hs + bs 1q<0 + bd 1q>0 ) + (w − ct )1q>0 − w1q<0

∂2 π (q, ns , nd , w, bs , bd ) ∂q 2 s

= −(rs + hs + gs + rl − bs 1q<0 )fs (ns − q) − (rs + bd 1q>0 )fd (nd + q) < 0

M

The profit function of the rail firm is

(F.1)

CE

PT

ED

πd (q, ns , nd , w, bs , bd ) = (rd + rl )E min{Xd , nd + q + Yd } − bs E min{(ns − q + Ys − Xs )+ , (−q)+ } −hd E(nd + q + Yd − Xd )+ + bd E min{(nd + q + Yd − Xd )+ , q + } −gd E(Xd − nd − q + Yd )+ + (w − ct )(−q)+ − wq + = (rd + hd + gd + rl − bd 1q>0 )E min{Xd − Yd , nd + q} + bs 1q<0 E min{Xs − Ys , ns − q} +(−hd + bs 1q<0 + bd 1q>0 )q +(w − ct )(−q)+ − wq + − bd 1q>0 E(nd + Yd − Xd )+ + bs 1q<0 E(ns + Ys − Xs )+ −(hd − bd 1q>0 )nd − bs 1q<0 ns − gd E(Xd − Yd ) + (rd + rl )EYd

AC

∂ π (q, ns , nd , w, bs , bd ) ∂q d

= (rd + hd + gd + rl − bd 1q>0 )F¯d (nd + q) − bs 1q<0 F¯s (ns − q) +(−hd + bs 1q<0 + bd 1q>0 ) + (w − ct )1q<0 − w1q>0 ∂2 π (q, ns , nd , w, bs , bd ) ∂q 2 d

= −(rd + hd + gd + rl − bd 1q>0 )fd (nd + q) − bs 1q<0 fs (ns − q) < 0

(F.2)

Appendix G. Proof of Theorem 3 The strategy space for each players is [−nd , ns ]. From Lemma 3, we know that qso weakly dominates all the other strategies for the liner firm and qdo for the rail firm. Hence, by eliminating weakly dominated strategies, there is a single strategy left for each player. The strategy profile {qso , qdo } is a Nash equilibrium. 39

ACCEPTED MANUSCRIPT

It is easy to check that under the equilibrium, the delivery quantity is q e = min{(qso )+ , (qdo )+ }− min{(qso )− , (qdo )− }. It indicates that the equilibrium quantity could only be qso , qdo or 0. The profit functions of both the liner firm and the rail firm are strictly concave. Hence, we can state that it is impossible to make one player better off without making another worse off. The Nash equilibrium is Pareto optimal. Appendix H. Proof of Theorem 4

CR IP T

The system profit under coordination is

Π(q o , ns , nd ) = (rs + hs + gs + rl )E min{Xs − Ys , ns − q o } + (rs + rd + hd + gd + rl )E min{Xd − Yd , nd + q o } +(hs − hd )q o − ct |q o | − hs ns − hd nd − gs E(Xs − Ys ) − gd E(Xd − Yd ) +(rs + rl )EYs + (rs + rd + rl )EYd

AN US

The seaport’s profit function without empty container transport is

πs (0, ns , nd , w, bs , bd ) = (rs + hs + gs + rl )E min{Xs − Ys , ns } + rs E min{Xd − Yd , nd } −hs ns − gs E(Xs − Ys ) + (rs + rl )EYs + rs EYd The dry port’s profit function without empty container transport is

M

πd (0, ns , nd , w, bs , bd ) = (rd + hd + gd + rl )E min{Xd − Yd , nd } − hd nd − gd E(Xd − Yd ) + (rd + rl )EYd

ED

From above arguments, we can write the profit increment of the system from the coordination contract as follows:

AC

CE

PT

Π(q o , ns , nd ) − πs (0, ns , nd , w, bs , bd ) − πd (0, ns , nd , w, bs , bd ) = (rs + hs + gs + rl )E min{Xs − Ys , ns − q o } + (rs + rd + hd + gd + rl )E min{Xd − Yd , nd + q o } −(rs + hs + gs + rl )E min{Xs − Ys , ns } − (rs + rd + hd + gd + rl )E min{Xd − Yd , nd } +(hs − hd )q o − ct |q o | R n −qo = (rs + hs + gs + rl )[(ns − q o )F¯s (ns − q o ) − ns F¯s (ns ) + nss zfs (z)dz] R n +qo +(rs + rd + hd + gd + rl )[(nd + q o )F¯d (nd + q o ) − nd F¯d (nd ) + ndd zfd (z)dz] +(hs − hd )q o − ct |q o | R n −qo R n +qo = (rs + hs + gs + rl ) nss (z − ns )fs (z)dz + (rs + rd + hd + gd + rl ) ndd (z − nd )fd (z)dz +[−(rs + hs + gs + rl )F¯s (ns − q o ) + (rs + rd + hd + gd + rl )F¯d (nd + q o ) + hs − hd ]q o − ct |q o |

40

ACCEPTED MANUSCRIPT

The seaport’s profit increment is

AN US

The marginal profits of the seaport and the dry port are

CR IP T

πs (q o , ns , nd , w, bs , bd ) − πs (0, ns , nd , w, bs , bd ) = (rs + hs + gs + rl − bs 1qo <0 )E min{Xs − Ys , ns − q o } + (rs + bd 1qo >0 )E min{Xd − Yd , nd + q o } −(−hs + bs 1qo <0 + bd 1qo >0 )q o + wq o − ct (q o )+ + (rs + bd 1qo >0 )E(nd + Yd − Xd )+ −bs 1qo <0 E(ns + Ys − Xs )+ − (hs − bs 1qo <0 )ns − bd 1qo >0 nd −gs E(Xs − Ys ) + (rs + rl )EYs + rs EYd −[(rs + hs + gs + rl )E min{Xs − Ys , ns } − hs ns − gs E(Xs − Ys ) + (rs + rl )EYs + rs EYd ] = (hs − bs 1qo <0 − bd 1qo >0 )q o + wq o − ct (q o )+ +(rs + hs + gs + rl − bs 1qo <0 )(E min{Xs − Ys , ns − q o } − E min{Xs − Ys , ns }) +(rs + bd 1qo >0 )(E min{Xd − Yd , nd + q o } − E min{Xd − Yd , nd }) R n −qo R n +qo = (rs + hs + gs + rl − bs 1qo <0 ) nss (z − ns )fs (z)dz + (rs + bd 1qo >0 ) ndd (z − nd )fd (z)dz +[−(rs + hs + gs + rl − bs 1qo <0 )F¯s (ns − q o ) + (rs + bd 1qo >0 )F¯d (nd + q o ) +hs + w − bs 1qo <0 − bd 1qo >0 ]q o − ct (q o )+ ∂πs (q,ns ,nd ,w,bs ,bd ) ∂q

= −(rs + hs + gs + rl − bs 1q<0 )F¯s (ns − q) + (rs + bd 1q>0 )F¯d (nd + q) +h  s − bs 1q<0 − bd 1q>0 + (w − ct )1q>0 + w1q<0 −(rs + hs + gs + rl )F¯s (ns − q) + (rs + bd )F¯d (nd + q) + hs − bd + (w − ct ), q > 0, = −(rs + hs + gs + rl − bs )F¯s (ns − q) + rs F¯d (nd + q) + hs − bs + w, q < 0,

M

and ∂πd (q,ns ,nd ,w,bs ,bd ) ∂q

PT

ED

= (rd + rl )F¯d (nd + q) + bs 1q<0 Fs (ns − q) − hd Fd (nd + q) + bd 1q>0 Fd (nd + q) +gd F¯d (nd + q) − (w − ct )1q<0 − w1q>0 = (rd + hd + gd + rl − bd 1q>0 )F¯d (nd + q) − bs 1q<0 F¯s (ns − q) −h  d + bd 1q>0 + bs 1q<0 − (w − ct )1q<0 − w1q>0 (rd + hd + gd + rl − bd )F¯d (nd + q) − hd + bd − w, q > 0, = ¯ ¯ (rd + hd + gd + rl )Fd (nd + q) − bs Fs (ns − q) − hd + bs − (w − ct ), q < 0.

CE

Case A: q o = ns .

AC

Π(q o , ns , nd ) − πs (0, ns , nd , w, bs , bd ) − πd (0, ns , nd , w, bs , bd ) R n −qo R n +qo = (rs + hs + gs + rl ) nss (z − ns )fs (z)dz + (rs + rd + hd + gd + rl ) ndd (z − nd )fd (z)dz +[−(rs + hs + gs + rl )F¯s (ns − q o ) + (rs + rd + hd + gd + rl )F¯d (nd + q o )]q o + (hs − hd )q o − ct |q o | Rn R n +n = (rs + hs + gs + rl ) 0 s (ns − z)fs (z)dz + (rs + rd + hd + gd + rl ) nds d (z − nd )fd (z)dz +[−(rs + hs + gs + rl )F¯s (0) + (rs + rd + hd + gd + rl )F¯d (ns + nd ) + hs − hd − ct ]ns

41

ACCEPTED MANUSCRIPT

The condition for voluntary compliance of the two players is ( ∂πs (q o ,ns ,nd ,w,bs ,bd ) ≥0 ∂q ∂πd (q o ,ns ,nd ,w,bs ,bd ) ≥0 ∂q ⇒ (rs + hs + gs + rl )F¯s (0) − rs F¯d (nd + ns ) − hs + ct ≤ w − bd Fd (nd + ns ) ≤ (rd + hd + gd + rl )F¯d (nd + ns ) − hd

CR IP T

πs (q o , ns , nd , w, bs , bd ) − πs (0, ns , nd , w, bs , bd ) R n −qo R n +qo = (rs + hs + gs + rl − bs 1q<0 ) nss (z − ns )fs (z)dz + (rs + bd 1q>0 ) ndd (z − nd )fd (z)dz +[−(rs + hs + gs + rl − bs 1q<0 )F¯s (ns − q o ) + (rs + bd 1q>0 )F¯d (nd + q o ) +hs + w − bs 1qo <0 − bd 1qo >0 ]q o − ct (q o )+ R n +n R0 = (rs + hs + gs + rl ) ns (z − ns )fs (z)dz + (rs + bd ) ndd s (z − nd )fd (z)dz +[−(rs + hs + gs + rl )F¯s (0) + (rs + bd )F¯d (nd + ns ) + hs − ct + w − bd ]q o

AN US

Given bd , the liner profit increment is increasing in w. Let w = (rd + hd + gd + rl − bd )F¯d (nd + ns ) − (hd − bd ), which is the profit upper bound for the liner firm to achieve. πs (q o , ns , nd , w, bs , bd ) − πs (0, ns , nd , w, bs , bd ) R0 R n +n = (rs + hs + gs + rl ) ns (z − ns )fs (z)dz + (rs + bd ) ndd s (z − nd )fd (z)dz +[−(rs + hs + gs + rl )F¯s (0) + (rs + rd + hd + gd + rl )F¯d (nd + ns ) + hs − hd − ct ]ns

ED

M

It is increasing in bd . From the second order derivatives of πs and πd , that is (F.1) and (F.2), we know that the concavity of the profit functions of both firms preserves when 0 ≤ bs ≤ rs + hs + gs + rl and −rs ≤ bd ≤ rd + hd + gd + rl . The upper bound is attained when bd = rd + hd + gd + rl then w = rd + gd + rl . The seaport earns the entire system profit increment. The lower bound is attained when bd = −rs and w = (rs + hs + gs + rl )F¯s (0) − rs − hs + ct .

CE

Case B: q o = qˆ. ( o

PT

πs (q o , ns , nd , w, bs , bRd ) − πs (0, ns , nd , w, bs , bd ) n = (rs + hs + gs + rl ) 0 s (ns − z)fs (z)dz = (rs + hs + gs + rl )[Is (0, ns ) − Is (q o , ns ) − q o Fs (ns − q o )] ∂πs (q ,ns ,nd ,w,bs ,bd ) ∂q ∂πd (q o ,ns ,nd ,w,bs ,bd ) ∂q

AC

=0 =0 ¯ ⇒ (rs + hs + gs + rl )Fs (ns − qˆ) − rs F¯d (nd + qˆ) − hs + ct = w − bd Fd (nd + qˆ) = (rd + hd + gd + rl )F¯d (nd + qˆ) − hd

Π(q o , ns , nd ) − πs (0, ns , nd , w, bs , bd ) − πd (0, ns , nd , w, bs , bd ) R n −ˆq R n +ˆq = (rs + hs + gs + rl ) nss (z − ns )fs (z)dz + (rs + rd + hd + gd + rl ) ndd (z − nd )fd (z)dz πs (q o , ns , nd , w, bs , bd ) − πs (0, ns , nd , w, bs , bd ) R n −qo R n +qo = (rs + hs + gs + rl ) nss (z − ns )fs (z)dz + (rs + bd ) ndd (z − nd )fd (z)dz +[−(rs + hs + gs + rl )F¯s (ns − q o ) + (rs + bd )F¯d (nd + q o ) + hs + w − bd − ct ]q o R n −qo R n +qo = (rs + hs + gs + rl ) nss (z − ns )fs (z)dz + (rs + bd ) ndd (z − nd )fd (z)dz 42

ACCEPTED MANUSCRIPT

R n −qo The seaport’s profit attains its maximum (rs + hs + gs + rl ) nss (z − ns )fs (z)dz + (rs + R n +˜q rd + hd + gd + rl ) ndd (z − nd )fd (z)dz when bd = rd + hd + gd + rl and w = rd + gd + rl , and R n −qo minimum (rs + hs + gs + rl ) nss (z − ns )fs (z)dz when bd = −rs and w = (rs + hs + gs + rl )F¯s (ns − qˆ) − rs − hs + ct .

CR IP T

Case C: q o = 0.  ∂+ πs (qo ,ns ,nd ,w,bs ,bd ) ≤0  ∂q    ∂+ πd (qo ,ns ,nd ,w,bs ,bd ) ≤ 0 ∂q ∂− πs (q o ,ns ,nd ,w,bs ,bd )  ≥0    ∂− πd (qo ,n∂q s ,nd ,w,bs ,bd ) ≥0 ∂q ¯ ⇒ (rd + hd + gd + rl )Fd (nd ) − hd ≤ w − bd Fd (nd ) ≤ (rs + hs + gs + rl )F¯s (ns ) − rs F¯d (nd ) − hs + ct (rs + hs + gs + rl )F¯s (ns ) − rs F¯d (nd ) − hs ≤ w − bs Fs (ns ) ≤ (rd + hd + gd + rl )F¯d (nd ) − hd + ct

AN US

Case D: q o = q˜. ( o

∂πs (q ,ns ,nd ,w,bs ,bd ) ∂q ∂πd (q o ,ns ,nd ,w,bs ,bd ) ∂q

=0 =0 ¯ ⇒ (rs + hs + gs + rl )Fs (ns − q˜) − rs F¯d (nd + q˜) − hs = w − bs Fs (ns − q˜) = (rd + hd + gd + rl )F¯d (nd + q˜) − hd + ct

M

Π(q o , ns , nd ) − πs (0, ns , nd , w, bs , bd ) − πd (0, ns , nd , w, bs , bd ) R n −˜q R n +˜q = (rs + hs + gs + rl ) nss (z − ns )fs (z)dz + (rs + rd + hd + gd + rl ) ndd (z − nd )fd (z)dz

CE

PT

ED

πs (q o , ns , nd , w, bs , bd ) − πs (0, ns , nd , w, bs , bd ) R n −qo R n +qo = (rs + hs + gs + rl − bs ) nss (z − ns )fs (z)dz + rs ndd (z − nd )fd (z)dz +[−(rs + hs + gs + rl − bs )F¯s (ns − q o ) + rs F¯d (nd + q o ) + hs + w − bs ]q o R n −qo R n +˜q = (rs + hs + gs + rl − bs ) nss (z − ns )fs (z)dz + rs ndd (z − nd )fd (z)dz R n −qo R n +˜q The seaport’s profit attains its maximum (rs +hs +gs +rl ) nss (z−ns )fs (z)dz+rs ndd (z− nd )fd (z)dz when bs = 0 and w = (rd + hd + gd + rl )F¯d (nd + q˜) − hd + ct , and minimum R n +˜q rs ndd (z − nd )fd (z)dz when bs = rs + hs + gs + rl and w = rs + gs + rl − rs F¯d (nd + q˜). Case E: q o = −nd .

AC

Π(q o , ns , nd ) − πs (0, ns , nd , w, bs , bd ) − πd (0, ns , nd , w, bs , bd ) R n −qo R n +qo = (rs + hs + gs + rl ) nss (z − ns )fs (z)dz + (rs + rd + hd + gd + rl ) ndd (z − nd )fd (z)dz +[−(rs + hs + gs + rl )F¯s (ns − q o ) + (rs + rd + hd + gd + rl )F¯d (nd + q o ) + hs − hd ]q o − ct |q o | R n +n R0 = (rs + hs + gs + rl ) nss d (z − ns )fs (z)dz + (rs + rd + hd + gd + rl ) nd (z − nd )fd (z)dz −[−(rs + hs + gs + rl )F¯s (ns + nd ) + (rs + rd + hd + gd + rl )F¯d (0) + hs − hd + ct ]nd

43

ACCEPTED MANUSCRIPT

(

∂πs (q o ,ns ,nd ,w,bs ,bd ) ∂q ∂πd (q o ,ns ,nd ,w,bs ,bd ) ∂q

≤0 ≤0

⇒ (rd + hd + gd + rl )F¯d (0) − hd + ct ≤ w − bs Fs (ns + nd ) ≤ (rs + hs + gs + rl )F¯s (ns + nd ) − rs F¯d (0) − hs The seaport’s profit increment is

CR IP T

πs (q o , ns , nd , w, bs , bd ) − πs (0, ns , nd , w, bs , bd ) R0 R n +n = (rs + hs + gs + rl − bs ) nss d (z − ns )fs (z)dz + rs nd (z − nd )fd (z)dz +[(rs + hs + gs + rl − bs )F¯s (ns + nd ) − rs F¯d (0) − hs − w + bs ]nd

Given bs , the profit increment is decreasing in w. Set w to its lower bound, we get

AN US

πs (q o , ns , nd , w, bs , bd ) − πs (0, ns , nd , w, bs , bd ) R0 R n +n = (rs + hs + gs + rl − bs ) nss d (z − ns )fs (z)dz + rs nd (z − nd )fd (z)dz +[(rs + hs + gs + rl )F¯s (ns + nd ) − (rs + rd + hd + gd + rl )F¯d (0) − hs + hd − ct ]nd It is decreasing in bs . To gain the upper bound of the profit increment, we set bs to 0.

M

πs (q o , ns , nd , w, bs , bd ) − πs (0, ns , nd , w, bs , bd ) R n +n R0 = (rs + hs + gs + rl ) nss d (z − ns )fs (z)dz + rs nd (z − nd )fd (z)dz +[(rs + hs + gs + rl )F¯s (ns + nd ) − (rs + rd + hd + gd + rl )F¯d (0) − hs + hd − ct ]nd

ED

πd (q o , ns , nd , w, bs , bRd ) − πd (0, ns , nd , w, bs , bd ) n = (rd + hd + gd + rl ) 0 d (nd − z)fd (z)dz = (rd + hd + gd + rl )[Id (0, nd ) − Id (q o , nd ) + q o Fd (nd + q o )].

AC

CE

PT

The seaport attains its lower bound of the profit increment R 0when bs = rs + hs + gs + rl and w = (rs +hs +gs +rl )−rs F¯d (0)−hs . The lower bound is rs nd (z −nd )fd (z)dz = rs |∆Id (q o )|.

44