The role of consignees in empty container management

Transcription

1 The role of consignees in empty container management B. Legros, Y. Bouchery, J.C. Fransoo Beta Working Paper series 517 BETA publicatie WP 517 (working paper) ISBN ISSN NUR Einhoven October 2016

2 The role of consignees in empty container management Benjamin Legros 1 Yann Bouchery 2 Jan Fransoo 3 1 Laboratoire Genie Inustriel, CentraleSupélec, Université Paris-Saclay, Grane Voie es Vignes, Chatenay-Malabry, France 2 Ecole e Management e Normanie, Laboratoire Métis, 30 Rue e Richelieu, Le Havre, France 3 Einhoven University of Technology, School of Inustrial Engineering P.O. Box 513, 5600 MB Einhoven, the Netherlans benjamin.legros@centraliens.net ybouchery@em-normanie.fr j.c.fransoo@tue.nl Abstract Nowaays, the majority of consumer goos is transporte into a maritime container uring at least one stage of the journey. Besies the many avantages of containerization, the management of empty containers is a key issue responsible for costly repositioning operations. This article investigates the potential for consignees to manage an inventory of empty containers at their location so as to enable irect reuse of these containers by shippers locate in the surrounings. Our moel aims at minimizing inventory holing an repositioning costs. These costs epen on the age of the container in the system in practice. We moel the problem as a Markov ecision process using the waiting time of the olest container as a ecision variable. Next, we use the value iteration technique to prove that a threshol policy in the age of the olest container in stock is optimal. This paper, to the best of our knowlege, is the first to provie a proof for the optimality of a threshol policy base on the age of the olest prouct in the system. We erive close-form formulas for the system performance uner the optimal policy. This allows us to compute numerically the optimal threshol an to erive explicit expressions of the optimal threshol in asymptotic regimes. We next analyze the impact of this proactive management of empty containers by the consignees on the level of irect container reuse. We show that this practice is very promising to enable a high level of irect reuse, but we also highlight that the consignees have little interest in implementing such a solution in the current setting of the container supply chain. We consequently propose remeies illustrate via a series of insights. Keywors: Empty container management, Markov ecision process, optimal inventory policy, value iteration technique, threshol. 1 Introuction Containerization is the tren towars the eployment of a stanar an eicate international transport system for containers from oor-to-oor. This inclues purpose-built containerships, eicate eep-sea terminals with special hanling equipment an eicate intermoal infrastructure in the hinterlan such as inlan terminals. We refer to Levinson (2010) for a historical perspective on containerization. Containerization has shape global supply chains by proviing reliable, low cost an secure service for international trae. Container-base trae has expene from 50 million Twenty-foot Equivalent Units (TEUs) in 1996 to 180 million TEUs in 2015 (Trae an Development (2016)), such that containerize cargo now represents more than half the value of all international seaborne trae (Trae an Development (2016)). As a result,

3 many authors consier ocean container transport as critical for global supply chain performance (Fransoo an Lee (2013)). As an example, Hausman et al. (2013) emonstrate the impact of logistics performance on trae by analyzing the time an costs of importing an exporting a container for 80 countries. Container transport has many avantages incluing stanarization, ease of hanling, protection against amage an security. However, empty container movement is often consiere as an important isavantage of containerization. Inee, once emptie at estination, the container often nees to be repositione to be fille in again. Empty container repositioning occurs at the regional level within small geographical areas, but also at the global level between major seaports ue to global trae imbalance (Boile et al. (2008)). For instance, the review of maritime transport publishe by the Unite Nations (2011) highlights that the costs of seaborne empty container repositioning was estimate to $20 billion in 2009, while the costs of empty container repositioning in the hinterlan was aroun $10 billion for the same year. Globally, empty container repositioning accounte for 19% of the global inustry income in 2009 (Unite Nations (2011)). We refer to e.g. Cheung an Chen (1998),Li et al. (2004), Olivo et al. (2005), Li et al. (2007), Lam et al. (2007), Erera et al. (2009), Long et al. (2015) for some stuies on the global repositioning problem. For regional repositioning, Braekers et al. (2011) estimate that empty containers account for 40% to 50% of the regional movements. In this article, we focus on empty container management in the hinterlan of a eep-sea port, i.e., at the regional level. Besies the cost of moving empty boxes, empty container repositioning in the hinterlan is responsible for many problems such as congestion in the port area an at eep-sea terminals, air pollution an accients. As a consequence, a lot of research has been evote to this issue. We refer to Dejax an Crainic (1987) for a review of early works in this fiel an we refer to Braekers et al. (2011) an Song an Dong (2015) for recent overviews. The main ecisions consiere eal with empty epots location (see e.g., Boile et al. (2008), Lei an Church (2011), Mittal et al. (2013)), container fleet sizing (see e.g., Du an Hall (1997)), inventory management for empty containers (see e.g., Yun et al. (2011)) an empty container routing (Jula et al. (2006), Deia et al. (2008), Furió et al. (2013)). Most hinterlan regions have both import an export flows. These flows are not balance in many cases an often very strongly imbalance. Let consier a global import hinterlan, i.e., a hinterlan for which the import flows are greater than the export flows. Some containers will nee to be back empty to the port, for global repositioning. However, some others will be reloae with export cargo in the hinterlan. For these containers, although inefficient, the common practice consists of having first the empty container being shippe back to the port, before being sent to another shipper where it is reloae. This leas to situations such as the one escribe by Jula et al. (2006) for the ports of Los Angeles/Long Beach. In 2000, more than a million empty containers became available in the hinterlan at local consignees sites an were move back empty to container terminals at one of the two ports. Meanwhile, 550,000 empty containers were transporte from the two ports to local shippers. This situation appears in many major ports worlwie an leas to unnecessary movements of empty containers. Besies the financial impacts associate with this practice, unnecessary movements of empty containers lea to negative societal impacts such as pollution, congestion an accients. 2

4 Braekers et al. (2011) highlight six strategies to reuce the unnecessary movements of empty containers in the hinterlan, i.e., the use of inlan container epots, street-turns, container substitution, container leasing, folable containers an Internet-base systems. We refer to the Appenix 1 in Braekers et al. (2011) for a etaile escription of these six strategies. This article focuses on the street-turn strategy as this one is often consiere as one of the most efficient strategies for empty container management in the hinterlan. The iea behin a street-turn is very basic an consists of shipping the empty container irectly from the consignee to the shipper without passing by a terminal. The avantages of the street-turn strategy are numerous. The empty movements are reuce an accoringly repositioning costs ecrease. Similarly, the reuction of empty movements may imply a ecrease in the number roa accients. Each street-turn also avois two movements to an from the terminal, reucing congestion. Finally, empty container eman from the shippers can be met sooner, increasing the container utilization rate which positively impacts container fleet sizing (Jula et al. (2006), Dong an Song (2012)). Even if the avantages of the street-turn strategy appear to be important for all parties involve, Lei an Church (2011) highlight that street-turns are only use 10% of the time in the hinterlan of Los Angeles/Long Beach. Ientically, Wolff et al. (2007) foun that the share of street-turns was in a range of 5-10% in the hinterlan of the port of Hamburg. There are multiple explanations for the limite use of street-turns. First, operational limitations inclues location an timing mismatch between the offer of empty containers at the consignees an the eman at the shippers, container type an ownership mismatch, lack of information sharing between consignees an shippers as well as the limite free time for container use by consignees an shippers (before incurring etention fees). Secon, institutional barriers inclue the lack of proceure for such a irect interchange between a consignee an a shipper as well as inspection an paperwork issues. Finally, insurance issues also contribute to limit the generalization of street-turns (Jula et al. (2006)). In this article, we aim at better unerstaning the operational limitations to street-turns. We especially focus on location an timing mismatch. This is in line with Song an Dong (2015) who consier that trae imbalance, ynamic operations an uncertainties are the critical factors explaining the high level of empty movements. Containers are generally controlle by the shipping lines who own an/or lease a pool of containers. As a consequence, the street-turn strategy is traitionally stuie from a shipping line perspective as in Jula et al. (2006), Deia et al. (2008) an Furió et al. (2013). As notice by Braekers et al. (2011), the consignee is generally not recognize as a player involve in empty container management at the regional level, an therefore, empty containers that becomes available at a consignee s site are irectly move back to the port or to an empty epot. Inee, managers often consier empty container movements as a necessary evil. This article analyzes the potential for consignees to become proactive in empty container management as we consier that they are at the best place to hanle the container location an timing mismatch issues. Matching empty containers in the hinterlan woul inee allow consignees to lower their empty container repositioning costs. Ocean container transport in general has eserve a lot of attention from the transport an maritime economics communities. Many relevant an inspiring results have been generate. However, the transport an maritime economics literature oes not take the inventory perspective into account, an hence their 3

5 research paraigms cannot be use to tackle the problem we consier here. We moel the problem as a Markov ecision process an use the age of the olest container as a ecision variable. This Operations Management (OM) technique allows us to moel the inventory of empty containers at a consignee s location. Examples of successful applications of OM in maritime transportation inclue container scheuling from a shipping line perspective (Choi et al. (2012)), ocean freight service purchasing by shippers (Lee et al. (2015)) an container inspections by customs (Bakshi an Gans (2010), Bakshi et al. (2011)). Several authors claim that ientifying options for street-turns is a challenging task (see e.g., Braekers et al. (2011)). We firmly believe that OM techniques will contribute to a better unerstaning of the opportunities an impeiments regaring street-turn strategies. In this article, we investigate the potential for consignees to proactively manage empty containers at their location to enhance the feasibility of street-turns. From our knowlege, this article is the first one to investigate this option. Our contribution is twofol. First, we ientify the optimal inventory policy as a threshol policy in the age of the olest container. To the best of our knowlege, this article is the first to prove the optimality of a threshol structure base on the time spent in the system using the value iteration metho. Next, we erive close-form expressions of the performance measures. This allows us to compute numerically the optimal threshol. Moreover, using an asymptotic analysis we erive an evaluate closeform expressions of the threshol. Secon, we apply the moel to several examples of inventory holing cost function of practical interest, an we erive a series of insights from the results obtaine. We show that the proactive management of empty containers by consignees enable reaching a high level of street-turns in many cases. However, we show show that the consignees have little incentive to manage empty containers as the cost reuction obtaine is often small ue to etention fees. We investigate potential remeies such as an increase in the etention free perio, the possibility for consignees to be exempte from etention fees for a small number of their containers, an a change in the etention fees structure an purpose. We organize the rest of the article as follows. The setting we consier is escribe in Section 2. The policy analysis, incluing the proof of the optimal policy an the performance analysis is evelope in Section 3. Section 4 is evote to the analysis of special cases of inventory holing cost functions. This analysis enables us to stuy the behavior of the optimal inventory policy an the performance of this policy on the level of street-turns, in orer to erive a series of practical insights. Finally, Section 5 is evote to the conclusion an to future research irections. All proofs are given in the appenix at the en of the article. 2 Setting Container transportation involves two main types of agreements between the shipping line an the consignee (respectively the consignor, epening on incoterms). Uner carrier haulage, the shipping line provies oor-to-oor services, i.e., the shipping line takes responsibility of container transportation from port-toport an in the hinterlan. Uner merchant haulage, the shipping line only takes care of port-to-port transportation. Inlan transportation is controlle by the consignee (resp. consignor) who often contracts a logistics service provier. The shipping line charges etention an emurrage fees for the use of her container in the hinterlan. Merchant haulage agreements are quite evelope. For instance, Notteboom (2009) states that the average share of merchant haulage is 60% for the port of Rotteram an 75% for the 4

6 port of Antwerp. Similarly, Sterzik (2013) states that the average share of merchant haulage is about 70% in Europe. Accoringly, we focus on merchant haulage in what follows an we assume that the consignee is responsible for container transportation in the hinterlan. This context is particularly relevant to stuy the role of consignees in empty container management. After the laen container is release at the eep-sea port, the consignee arranges transportation an receives the container. Then the container is unloae an nees to be sent back to the shipping line. In the vast majority of the cases in practice, the consignee ecies to sen the empty container back irectly after being unloae to the eep-sea terminal or to an empty epot. In this article, we investigate the option of keeping empties in inventory at the consignee s location until a shipper neeing an empty container is ientifie, to achieve a street-turn. We exclue institutional barriers against street-turns as well as insurance issues from the analysis. We focus on a given type of container (e.g., 40 ft. ry containers) as container size is stanarize. This leas to exclue container type mismatch from the analysis. Finally, ue to concentration in the shipping line inustry (Fransoo an Lee (2013)) an alliances (Agarwal an Ergun (2010)), we also exclue ownership mismatch from the analysis. Uner merchant haulage, the shipping lines inirectly control their containers in the hinterlan by charging etention an emurrage fees. Demurrage fees are incurre when the container stays at the eep-sea terminal, an etention fees are charge when the container is in the hinterlan, until being shippe back to the shipping line. Shipping lines want to get their containers back as quickly as possible, so they charge high etention an emurrage fees. As a consequence, consignees are sensitive to etention an emurrage fees an ten to negotiate etention an emurrage free perios. This leas to a complex structure in practice, with a free perio, an several levels of fees. As an example, let us focus on a 40 ft. ry container importe to the port of Rotteram by Maersk an assume that truck transportation is use to eliver the container to the consignee. The official etention fees accoring to Maersk (2016) inclue a free perio of 3 ays, then the rate is e55/ay from ay 4 to ay 7 an finally the rate is e85/ay after 7 ays. In our iscussions with consignees across Europe (Netherlans, France an Sween), etention fees are always mentione as one of the main barriers against street-turn strategies as the consignees feel that they o not have enough time to ientify an export match before incurring high etention costs. Our moel inclues etention fees an we are consequently able to analyze the impact of etention fees on the level of street-turns. We consier the regional empty container management problem from the consignee s perspective in isolation from the problem of managing the cargo insie the containers. Inee, the inventory management ecisions traitionally focus on the cargo an exclue the transportation packaging units. As a first attempt in analyzing the management of empty containers from the consignee perspective, we aim at focusing on moels that o not affect the management of the cargo itself. The supply of empty containers is consequently not consiere as a ecision variable in our moel an we consier this one as stochastic. We assume that the arrival process of empty containers is Poisson with parameter λ. We assume that transportation costs in the hinterlan are incurre by the consignee, i.e., the consignee bears the costs for shipping the laen container from the eep-sea terminal to the consignee s location (in accorance with the merchant haulage setting) as well as the costs for repositioning the empty container. 5

7 Two options are available. Either the empty container is shippe back to the shipping line (either to the eep-sea terminal or to an empty epot epening on the cheapest available option), an incurs a cost of c sl monetary units per empty container, or the consignee manages to ientify a shipper in nee of an empty container in a nearby location, an incurs a cost of c st monetary units per empty container. We assume that the transportation costs an time from the consignee to the shipper are lower than the transportation costs an time from the consignee to the shipping line (otherwise, the consignee has no interest in sening any empty container to the shipper). We consequently efine the repositioning costs as c s = c sl c st (c s > 0). In case of a street-turn, the shipper will receive an empty container for free (as the consignee is paying for transportation) an quicker than a container obtaine from the shipping line. We accoringly assume that the shipper will always accept the street-turn. Moreover, we assume that all the shippers of interest for the consignee are locate in the same zone, such that transportation costs an time from the consignee to each feasible shippers are similar. We further assume that the aggregate eman for empty containers is stochastic an follows an exponential istribution with rate µ. Note that we refer to a single shipper for clarity reasons. The objective for the consignee is to fin the optimal empty containers inventory policy which minimizes the long-run expecte costs. In case an empty container is kept in inventory, the consignee incurs linear inventory holing costs ue to physical storage costs as well as etention costs if the etention free perio is over. We aggregate these costs an we refer to them as inventory holing costs for simplicity. The inventory holing costs epen on the actual time spent by a container in the hinterlan. The cost structure is not necessarily a linear function ue to etention fees but it is often a more complex convex function. Although a state escription base on the number of containers in the hinterlan is common in Markov chain analysis, it oes not allow to evaluate the overall cost per time unit of a set of containers. The information on the age of each container shoul be ae in orer to completely escribe a given state of the system. This woul make the state escription very complex. Since we assume that the inventory holing cost per time unit is an increasing an convex function of time, the olest container in the hinterlan is also the most costly container. Hence, if a ecision to sen back a container to the shipping line shoul be taken, this ecision shoul be in priority taken for the olest container. In other wors, ue to the increasing structure of the inventory holing costs, it is optimal to apply a first-in-first-out policy for sening back containers. For the same reason, a first-in-first-out iscipline is optimal when a container can be sent to the shipper. Combining the Poisson arrival process with the first-in-first-out iscipline, the age of the olest container allows us to evaluate at any time the istribution of the containers as a function of their age. Therefore the age of the olest container is chosen as a ecision variable to etermine the optimal policy. We prove, in the setting consiere, that the optimal control policy is a threshol policy on the age of the olest empty container in inventory. We erive close-form formulas for the performance measures. This allows us to obtain the optimal threshol. For several examples of inventory holing cost functions, we stuy the impact of the parameters on the performances of empty container management an we erive some practical insights. 6

8 3 Policy Analysis In this section, we prove that the optimal control policy is a threshol policy on the age of the olest container. Next, we compute the performance measures uner this policy. 3.1 Optimal policy We consier the set of all non-preemptive non-anticipating first-in-first-out policies for sening back containers to the shipping line. At any point of time, we want to ecie for the olest container (if any) whether to keep it, or to sen it back to the shipping line. Whenever a match is possible it is optimal to realize the match. Since interarrival times of empty containers are exponential, the information on the olest container allows us to estimate the istribution of the queue length (Koole et al. (2012)). Therefore the queue length oes not nee to be another ecision variable because it completely epens on the age of the olest container. The objective function is compose by the inventory holing costs (incluing etention costs) an the costs of sening back a container to the shipping line. Concretely, the goal is to fin the optimal policy which minimizes the following weighte sum which represents the long-run expecte cost per container; E(C) = c s P s + E(D), (1) where the coefficient c s (c s 0) is the repositioning cost efine in Section 2 (Recall that c s is the ifference between the costs of sening back a container to the shipping line an the costs of shipping a container to the shipper), P s is the long-run expecte proportion of containers sent back to the shipping line an E(D) is the expecte inventory holing cost for a given container. We propose to formulate the problem as a Markov ecision process an next use the value iteration technique to prove the form of the optimal policy. We propose here a non-stanar efinition for the system states. We explicitly moel the age of the olest container, instea of the traitional moeling using the number of containers. This iea has been first propose by Koole et al. (2012) in orer to analyze queueing systems. The approach consists first of iscretizing the age of the olest container using successive exponential stages, each with rate γ, an next consists of reporting the waiting stage in the Markov process. The total number of stages require is not known beforehan. This is etermine by the eman from the shipper. Having large values of γ improves the approximation as it better represents the continuously elapsing time. As γ tens to infinity, this approximate setup converges to the original one, which in turn leas to an exact analysis. Let us enote by x a state of the system, where x 0. State x = 0 correspons to an empty inventory. States with x > 0 correspon to a situation where the olest container in the system has an age of x stages. We enote the corresponing transition rate from state x to state x by t x,x. Recall from Koole et al. (2012) ( ) ( ) h λ that the transition probabilities enote by q x,x h from a stage x to a stage x h, are q x,x h = γ λ+γ λ+γ 7

9 ( ) x an q x,0 = γ λ+γ for x > 0 an 0 h < x. Hence, for x, x 0, we may write λ, if x = 0, x = 1, γ, if x = x + 1, x > 0 t x,x = µq x,x h, if x = x h, x > 0, an 0 h x 0, otherwise, which correspons to arrival, eman an elapsing times. Let us enote by V n (x) the expecte cost over n steps, for n 0 an x 0. We pay a cost of c s per container sent back to the shipping line an let c(x) be the inventory holing cost function per container. We assume that c(x) is a general increasing an convex function of the age the container. The cost c(x) can therefore be use to moel both the holing cost of storage an the etention cost which can inclue a etention free perio. We choose to iscretize our continuous-time moel. This is possible because it is uniformizable (Section in Puterman (1994)). The uniformization is one using the maximal event rate λ + µ + γ, that we assume equal to 1. We enote by F the operator on the set of functions f from N to R efine by F (f(x)) = x q x,x h f(x h) for x > 0, an F (f(0)) = f(0) for x = 0. For n 0, we have V n+1 (0) = λw n (0) + (1 λ)v n (0), V n+1 (x) = γw n (x) + µ(f (V n (x)) + c(x)) + (1 γ µ)v n (x), for x > 0, with W n (x) = max(f (V n (x)) + c s + c(x), V n (x + 1)) if x 0. We assume that V 0 = W 0 = 0. For each n > 0 an every state x (x 0) there is a minimizing action: sen an empty container to the shipping line or keep all containers in the inventory. For fixe n (n > 0) we call this function: N {keep, sen}, a policy. As n tens to infinity, this policy converges to the average optimal policy, that is, the policy that minimizes the long-run expecte average costs (Puterman (1994)). In Theorem 1, we prove that the optimal policy at iteration n for sening back containers is of threshol type uner the conitions µ > λ, γ > µ an c(1) < c s. In Corollary 1, we show that in the long run, as γ tens to infinity, the optimal policy is also of threshol type without any restricting conitions on the system parameters. To the best of our knowlege, this paper is the first to prove the optimality of a threshol structure base on the time spent in a system using the value iteration technique. Most of continuous time Markov ecision processes use the number in the system as a ecision variable. This allows simple ecisions but may not be optimal in a setting where the cost function substantially changes over time. Koole et al. (2012) provie a metho to moel the waiting time of a customer in a queue to allow for ecisions base on the waiting but i not use the metho to theoretically prove structural properties of a policy. Another 8

10 numerical illustration of the metho can be foun in Koole et al. (2015). Theorem 1 For µ > λ, γ > µ, c(1) < c s, the optimal policy for sening back containers at iteration n is of threshol type. There exists a threshol on the number of stages at which an empty container is sent back to the shipping line. Corollary 1 As γ tens to infinity, the optimal long-run policy is a threshol policy on the age of the olest empty container. The optimal routing result in Theorem 1 an Corollary 1 allow us to efine more precisely the optimal rejection policy for empty containers. All containers are allowe to join the inventory, regarless of the system state. However, the system oes not allow containers to infinitely stay in the inventory. A container waiting in at the consignee since exactly τ time units is sent back to the shipping line. Note that this policy is very simple to implement in practice. 3.2 Performance Analysis We have proven in Section 3.1 that the optimal policy for the consignee is a threshol policy on the age of the empty containers. We evaluate here the performance of the system uner this policy. We approximate the eterministic uration before rejection by an Erlang ranom variable with n stages an rate γ per stage. We choose n an γ such that n γ converges to the eterministic uration before rejection, τ. = τ. This ensures that as n an γ go to infinity, this Erlang ranom variable We use the same state efinition as in Section 3.1 except that the total number of stages in boune. We enote by n the maximal number of stages. The transition structure is ientical to the one in 3.1. The only ifference is the transition from state x = n; a transition from n to a state n h can be cause not only by a µ-transition but also by a γ-transition which represents then a container sent back to the shipping line. In Theorem 2, we give close-form expressions for the probability of an empty system p 0 ( ), the proportion of containers sent back to the shipping line, P s, the expecte time spent by a container in the inventory, E(T ), the expecte number of container in the inventory, E(N), an the proportion of empty containers which has spent more than t in the inventory, P (T > t) with 0 t τ. These expressions are given as functions of the ratio a = λ/µ. The ratio a represents the import/export balance. Theorem 2 We have: p 0 ( ) = 1 a, (2) 1 ae τ(µ λ) P s = (1 a)e τ(µ λ), (3) 1 ae τ(µ λ) E(T ) = 1 e τ(µ λ) (1 + aτ(µ λ)), (4) µ(1 a)(1 ae τ(µ λ) ) 9

11 E(N) = a 1 e τ(µ λ) (1 + aτ(µ λ)), (5) (1 a)(1 ae τ(µ λ) ) where 1 (x A) is the inicator function of a subset A. P (T > t) = 1 t<τ e t(µ λ) ae τ(µ λ) 1 ae τ(µ λ), (6) Our system reuces to an M/M/1+D queue where the arrival process is generate by the arrival of empty containers an the service is ensure by the shipper. The analysis to compute the performance measures is base on the iscretization of the time spent by containers at the consignee s location. This metho is an alternative to the first analysis of the M/M/1+D queue realize by Baccelli an Hebuterne (1981). Cost function. In what follows, we erive the expression of the optimal cost function in the case where the inventory holing cost function is piecewise linear increasing (recall that this one inclues etention fees). This enables us to obtain the optimal threshol τ an the relate optimal performance measures. We enote by c k, the slope of the function on the intervals [t k 1, t k ) with the convention that t 0 = 0 for 1 k n. The cost function is convex, therefore c k is increasing in k. As a function of the time spent at the consignee, the cost function per container can be written as: c(t) = c 1 t1 t<t1 + (c 2 t t 1 (c 2 c 1 ))1 t1 t<t 2 + (c 3 t t 1 (c 2 c 1 ) t 2 (c 3 c 2 ))1 t2 t<t 3 + ) n 1 = 1 tn 1 t<t n (c n t t k (c k+1 c k ). n=1 k=1 Using the results of Theorem 2, we obtain the expecte cost per container; (1 τ a)e τ(µ λ) (1 a)e t(µ λ) E(C) = c s + c(τ) + µ 1 ae τ(µ λ) t=0 1 ae τ(µ λ) We enote by n the integer such that t n τ < t n +1, we thus have: τ t=0 n=1 ) n 1 n 1 tn 1 t<t n e (c t(µ λ) n t t k (c k+1 c k ) t = = c(τ)e τ(µ λ) µ λ Finally, we obtain: + τ + t=t n 1 (µ λ) 2 E(C) = c s (1 a)e τ(µ λ) 1 ae τ(µ λ) + 1 µ k=1 ) n 1 1 tn 1 t<t n (c n t t k (c k+1 c k ) t. n=1 n=1 tn ) n e (c t(µ λ) n +1t t k (c k+1 c k ) t k=1 t=t n 1 e t(µ λ) ( ) n c 1 + e tk(µ λ) (c k+1 c k ) c n +1e τ(µ λ). k=1 k=1 ( ) n 1 c n t t k (c k+1 c k ) t c 1 + n (e tk(µ λ) + at k (µ λ)e τ(µ λ) )(c k+1 c k ) c n +1e τ(µ λ) (1 + τa(µ λ)) k=1. (1 a)(1 ae τ(µ λ) ) k=1 10

12 Structural results. In proposition 1, we give the first an secon orer monotonicty properties of the main performance measures as a function of the control parameter τ. These properties will be use to evelop a metho to compute numerically the optimal threshol for a general cost function. Moreover, these results will also be use in Section 4 to better explain the behavior of the expecte cost. Proposition 1 For a > 0 an τ > 0, the following hols: 1. The proportion of containers sent back to the shipping line, P s, is strictly ecreasing an strictly convex in τ (the proportion of matches is thus strictly increasing an strictly concave in τ). 2. The expecte time spent at the consignee E(T ) an the expecte number of containers are strictly increasing in τ. 3. The proportion P (T > t) is strictly increasing in τ for t < τ. Note that the expecte time spent at the consignee as well as the expecte number of containers at the consignee are neither convex nor concave in τ for τ > 0. Note also that these monotonicity results apply for the the M/M/1+D queue. To the best of our knowlege these results can not be foun in the queueing literature. Optimal threshol. Using the monotonicity results, we evelop a metho to obtain the optimal threshol: 1. On each interval [t k, t k+1 ] we compute E(C) ; E(C) = e τ(µ λ) ( ( ) c (1 ae τ(µ λ) ) 2 s µ(1 a) 2 c 1 a + c n +1 (1 a)(1 + τaµ) + e τ(µ λ) (1 + τaµ(1 a) 2 ) ) + 1 k (e ti(µ λ) + t i (µ λ)e τ(µ λ) (1 2a))(c i+1 c i ). µ i=1 2. Either there exists a value for τ which is a zero of the above expression on the interval [t k, t k+1 ] or we choose between t k or t k+1 the value for τ which minimizes E(C) on the interval. The value which woul be a zero of E(C) is unique. 3. We obtain a countable set of possible values for τ. Moreover, this set of values is finite ue to the strict convexity of the cost function. Thus, as soon as increasing τ in a new interval only increases the expecte cost the search for the optimal threshol finishes in the previous interval. This metho to obtain the optimal threshol can be relate to other known problems in operations management like the optimal sizing of an inventory with iscounte prices in Wilson s moel. The optimal size in this moel is either at the extremity of a segment or a zero of the erivative of the expecte cost. Asymptotic results. Due to the strong imbalance which can appear in an import zone (a >> 1) or in an export zone (a << 1) it might be interesting to erive asymptotic expressions of the performance measures. f(a) We write f(a) g(a) to express that lim a0 a a g(a) = 1, for a 0 R. 0 11

13 In an import zone (a >> 1), we have: P s 1 1 a a, E(N) λτ, P (T > t) 1 ( t<τ 1 e (τ t)µ(1 a)). a a In an export zone (a << 1), we have: P s (1 + τλ)e τµ, a 0 E(N) a 0 a(1 e τµ (1 + τλ)), P (T > t) a 0 1 t<τ ( e tµ (1 + tλ) ). These expressions will be use in the next section to obtain approximations of the optimal threshol. 4 When shoul we sen back empty containers? In this section, we consier ifferent examples of inventory holing cost functions in orer to gain insights on when to sen back empty containers to the shipping line. We characterize situations for which the common practice that consists of sening back irectly all containers (referre to as the immeiate return policy) is optimal or not. We compare between the costs of our optimal policy to the costs obtaine uner the immeiate return policy. We also analyze how our policy affects the proportion of street-turns. 4.1 Constant inventory holing costs per time unit We consier here that the inventory holing cost per time unit is constant. This structure may apply if we exclue etention fees from the analysis. This will enable us assessing the impact of etention costs an complex etention costs structure on the performance of the system. The moel may inee also be use to assess the feasibility an effectiveness of implementing linear etention costs instea of the classical piecewise linear structure use. We enote by c u the inventory holing cost of an empty container per unit of time. The key insight from Proposition 2 below is that the ecision not to sen back containers immeiately to the shipping line only epens on the eman for empty containers; the flow of arriving containers oes not influence the ecision. More precisely, Proposition 2 proves that in case of a linear inventory holing cost function, our policy performs better than the immeiate return policy if an only if c s µ c u. In other cases, the immeiate return policy is optimal. Proposition 2 Two cases shoul be consiere. 1. If c s µ c u, it is optimal to sen back all empty containers to the shipping line (i.e., applying the immeiate return policy), 2. If c s µ > c u, there exists a unique finite threshol τ (τ > 0) which minimizes the expecte cost. This threshol is the unique solution of c s µ(1 a) 2 = c u (1 2a + a(1 a)µτ + a 2 e µ(1 a)τ ). (7) We now evaluate the impacts of c s an c u on the optimal policy. In proposition 3 an in corollary 2, we prove the impact of c s an c u on the optimal threshol τ an on the performance measures. Corollary 2 follows irectly from combining proposition 1 an proposition 3. 12

14 Proposition 3 The optimal threshol τ is increasing an concave in c s an ecreasing an convex in c u. Corollary 2 The proportion of matches, the expecte number of empty containers at the consignee an the optimal cost per container are increasing an concave in c s an ecreasing an convex in c u. Impact of c s. In Figure 1, we illustrate the optimal policy as a function of c s for a holing cost of e5 per ay for ifferent values of the arrival rate an an expecte eman, µ, of 1 container per ay. We take a = 1, a = 2 an a = 5 as examples. These values are representative of imbalance in major trae lanes. For instance, Theofanis an Boile (2009) report that trae imbalance for the transpacific lane increase from 50% (i.e., a = 2) to 67% (i.e., a = 3) from 2000 to In Figure 1(a), we compute the optimal threshol, i.e., the maximal uration of a container at the consignee before being sent back. In Figure 1(b), we give the relate optimal expecte number of containers in the inventory. In Figure 1(c), we compute the optimal cost per container. As a comparison we also give the cost per container if the immeiate return policy is implemente (note that in this case, the cost per container is equal to c s ). In Figure 1(), we evaluate the proportion of matches. This proportion is compute as the ratio of the expecte number of matches an the expecte eman from the shipper on a given perio of time. We inee consier that λ µ in our example so the maximum number of matches is limite by the eman for empty containers by the shipper. e e K e l l l l l l e e e e (a) Optimal threshol (b) Expecte number of containers in the inventory e K l l l / l el el el el l W l e l l l l l l l e e e e (c) Optimal cost per container () Proportion of matches Figure 1: Optimal policy analysis (µ = 1 container/ay, c u =e5/ay) We observe that: 1. As c s increases (i.e., as the extra cost for sening back a container to the shipping line increases), τ 13

15 increases. This result is proven in Proposition 3 an Corollary 2. It is quite straightforwar as an increase in c s implies an increase in the expecte benefit in case a match is realize. 2. As c s increases, the average number of container in inventory increases (as a consequence of the increase in τ). This result irectly follows from Proposition As c s increases, the absolute an relative profit obtaine when applying the optimal policy (as compare to the immeiate return policy) increases. For reference, within the Netherlans, the cost c s is about e100, which implies that the potential savings per container are from 16% for λ = 5 to 69% for λ = When omitting etention fees, the proportion of matches is very high as soon as c s µ > c u. This means that empty container management by consignees shows a very high potential for reucing unnecessary movements of empty containers in the hinterlan. Again, for the Dutch case with c s =e100, note that the number of matches well excees 80% even for λ = 1. However, if λ >> µ the ifference between the optimal cost per container an the cost in case of immeiate return is very low. This means that the consignees o not have strong incentives to manage empty containers at their location. 5. In Figure 1(b), we can notice that the curves cross each other. So, if c s is low the size of the inventory increases with λ an the opposite hols when c s is high. This results from the competition of two phenomenons. First, for low arrival rates a relatively high number of containers is require in inventory in orer to serve the relatively more volatile eman from the shipper. Secon, for high arrival rates (with relatively lower eman volatility per unit of time) a high number of containers in inventory may be require in the inventory in orer to reuce the flow of containers sent back to the shipping line with a cost c s. 6. In case the eman for empty containers is hel constant, the benefit of implementing the optimal policy ecreases as the arrival rate of containers increases. In the opposite, the proportion of matches increases as the arrival rate of empty containers increases. Impact of c u. We illustrate the impact of c u on the results with Figure 2. We set c s =e80/container an µ =1 container/ay. The results are compute as a function of c u for ifferent values of λ. Similarly to Figure 1, Figure 2(a) provies the optimal threshol in ays, Figure 2(b) illustrates the resulting average number of containers in inventory. Figure 2(c) illustrates the optimal cost per container an Figure 2() focuses on the resulting number of matches. We observe that: 1. As c u increases, the optimal threshol as well as the average number of containers in inventory ecreases (Figures 2(a) an 2(b)). This result is proven in Proposition 3 an Corollary The optimal cost per container is increasing in c u. Aitionally, ue to the concavity in c u (Corollary 2) the optimal cost per container is very sensitive to c u for low values of c u. 3. The proportion of matches is ecreasing in c u an the average number of matches is very sensitive to c u. Aitionally, for the parameter uner stuy, the average number of matches is almost linearly 14

16 e K l l l l l l e e e e (a) Optimal threshol (b) Expecte number of containers in the inventory e e K l el W e el l e el l l l l el l l l l / l l l e e e e (c) Optimal cost per container () Proportion of matches Figure 2: Optimal policy analysis (µ = 1 container/ay, c s =e80/container) eceasing in c u, especially for high values of λ (Figure 2()). This reveals that the asymptotic behavior is almost reache. This observation iffers from the convexity observe in Figure 2(a). One reason is that the proportion of matches is boune whereas the number of containers is not. Another reason is that the expecte number of containers involves both the first an the secon orer moments of the arrival an the eman process whereas the proportion of matches only involves the first moments. This explains also partly the stronger convexity of the number of containers. 4. As for Figure 1(b), we notice in Figure 2(b) that the curves cross each other. The reason is similar. If c u is low then the size of the inventory ecreases with λ an the opposite hols when c u is high. In case of large values for λ, the probability of receiving a eman from the shipper before receiving a new empty container is quite small, so only a small number of containers in inventory ensures a high level of matches. Otherwise, more containers in inventory are necessary to obtain the same level of matches. 5. Again, as the arrival of containers increases, the benefits of implementing the optimal policy ecreases an the proportion of matches increases. Asymptotic analysis. Even if Equation (7) can be easily solve numerically, it oes not lea to an explicit expression of the optimal threshol. We therefore aitionally provie explicit expressions of the optimal threshol in extreme cases of the import/export balance. These expressions are erive from Equation (7) using Taylor expansions an equivalent expressions. The following hols in the case c s µ > c u ; 15

17 As a tens to infinity (if the import/export balance is high), Equation (7) leas to τ 1 a λ ln ( cs µ c u ). As a tens to zero (if the arrival of containers is low or if the eman is high), Equation (7) leas to τ ( ) 1 cs µ 1. a 0 λ c u In Table 1, we evaluate the expecte cost obtaine using these two expressions of the threshol in comparison with the use of the optimal threshol for ifferent values of the arrival rate. In the secon an the thir column we give the optimal threshol an its relate expecte cost. We compute the relative ifference between the expecte cost obtaine with the two approximations an the optimal expecte cost by r= E(C) approximation E(C) optimal E(C) optimal. As expecte, Table 1 reveals that the first approximation is the best for high Table 1: Performance comparison (µ = 1, c s = 80, c u = 5) Optimal policy ( Approximation ) 1 ( Approximation ) 2 λ τ opt E(C) opt τ 1 = 1 λ ln cs µ E(C) c 1 r 1 τ 2 = 1 cs µ 1 E(C) u λ c 2 r 2 u % % % % % % % % % % % % % % % % % % % % % % % % arrival rate situations whereas the secon one is the best for low arrival rate situations. However, it is interesting to observe that uner low arrival rate situations the first approximation still performs well. The explanation is given by the result ln(1 + x) x 0 x. Hence, the secon expression foun in the case where a tens to zero is equivalent to the first one foun in the case where a tens to infinity. Note also that the worst egraation of the performance is observe with the secon approximation. 4.2 Impact of the etention free perio We assume here that the cost function is a simple one-step function; uring a first etention free perio no cost are encountere per container at the consignee. At the en of the etention free perio, the cost is assume to be infinite. This choice oes not correspon to a realistic cost moeling but this correspons to the behavior of a consignee that refuses to pay any etention fees, while neglecting the physical cost of keeping empty containers (the latter is often very low as the consignees often have some space available for staking few empties). This behavior is very common in practice accoring to the iscussions we ha with consignees in several European countries. In the setting we consier in this section, the optimal threshol is exactly equal to the uration of the etention free perio. 16

18 The expecte cost per container is therefore: E(C) = c s (1 a)e τ(µ λ) 1 ae τ(µ λ). In Figure 3, we present the impact of the uration of the etention free perio on the ifference in cost per container between the optimal policy an the immeiate return policy (Figure 3(a)) an on the proportion of matches (Figure 3(b)). Note that we take c s =e30/container an µ = 1 container/ay through this example. l W K l l l el el el el / l l l l l l l l l (a) Optimal cost per container (b) Proportion of matches Figure 3: Optimal policy analysis (µ = 1 container/ay, c s =e30/container) We observe that: 1. As expecte, the optimal cost per container is ecreasing an convex in the etention free perio, while the proportion of matches is increasing an concave in the etention free perio. This observation is proven in Corollary Empty container management by the consignees has a very high potential for reucing the unnecessary movements of empty containers in the hinterlan as soon as the resiual etention free perio at the consignee is high enough. 3. For high values of λ, the ifference between the optimal cost per container an the cost in case of immeiate return is very low. This means that the consignees o not have strong incentives to manage empty containers at their location in such a setting. This observation confirms the ones mae in Section 4.1 with constant inventory holing costs. 4. In the case of a small λ, the proportion of matches is strongly affecte by the etention free perio for small values of τ. This result irectly follows from Proposition 1. Corollary 3 gives the impact of the etention free perio on the optimal cost per container an the proportion of matches. The proof of corollary 3 irectly follows from proposition 1. Corollary 3 The following hols: 17

19 The optimal cost per container is ecreasing an convex in the etention free perio. The proportion of matches is increasing an concave in the etention free perio. 4.3 Managerial insights. The theoretical results an numerical tests presente in this article enable us to erive a series of insights on how to enhance empty container reuse in the hinterlan by focusing on street-turn strategies. We propose a new moel that enables consignees to optimally hanle a small amount of empties in orer to ientify possible export matches. The insights erive below aim at assessing the impacts of such proactive management of empty containers by consignees. Insight 1 The proactive management of empty containers by the consignees enables reaching a high level of street-turns in many cases. Street-turns are beneficial not only for the consignees (as a way to ecrease their costs) but also for the shipping lines (by increasing the utilization rate of the containers an reucing the cost of sening back empty containers to shippers). Street-turns aitionally enable reucing congestion, accients an pollution. Insight 1 is irectly erive from the observations mae about the level of matches in Sections 4.1 an 4.2. Insight 1 is very important as this shows that the new management practice stuie in this article may strongly help in optimizing container flows in the hinterlan. However, Insight 2 shows that the consignees may not be willing to investigate the iea of managing empties at their location unless they are incentivize to o so in the current setting of many hinterlans. Insight 2 In case of a strong imbalance in favor of import between import an export flows an short etention free perios, the consignees have little incentive to manage empty containers at their location as the ifference in cost as compare to the immeiate return policy is small. Insight 2 is erive from the observations mae in Sections 4.1 an 4.2 about the optimal costs per container as compare to the costs in case the immeiate return policy is implemente. This secon insight may be quite isappointing at first glance. Inee, combine with Insight 1, we face a very effective management practice to hanle the problem of reucing empty movements of containers in the hinterlan, but we show that this practice is not very likely to be implemente. However, our results help in ientifying how shipping lines may influence the behavior of the consignees an incentivize them to manage a small number of empty containers at their location. We consequently take the shipping line perspective in what follows. First, we assess if etention fees are efficient in controlling the time spent by containers in the hinterlan. We highlight in Section 2 that etention fees are often use as a way to ensure that containers are quickly shippe back to the shipping line. Now assume that we omit etention fees. Many shipping line companies woul be reluctant to implement such type of tariff as they woul be afrai of losing control of their containers, by incentivizing consignees to keep empties for a long time. The results we obtaine in Section 4.1 hol. Figure 1(a) shows that optimal threshol is quite low, an even very low for consignees with high eman. Moreover, Figure 2(a) shows that this optimal threshol is very sensitive to c u. This means that the consignees have little or no interest in keeping empty containers at their location as soon as they incur even 18