Real-time Container Transport Planning with Decision Trees based on Offline Obtained Optimal Solutions

Transcription

1 Real-time Container Transport Planning with Deision Trees based on Offline Obtained Optimal Solutions Bart van Riessen Eonometri Institute Erasmus Shool of Eonomis Erasmus University Rotterdam Rudy R. Negenborn Department of Marine and Transport Tehnology Delft University of Tehnology Rommert Dekker Eonometri Institute Erasmus Shool of Eonomis Erasmus University Rotterdam EI Marh

2 Abstrat Hinterland networks for ontainer transportation require planning methods in order to inrease effiieny and reliability of the inland road, rail and waterway onnetions. In this paper we aim to derive real-time deision rules for suitable alloations of ontainers to inland servies by analysing the solution struture of a entralised optimisation method used offline on histori data. The deision tree an be used in a deision support system (DSS) for instantaneously alloating inoming orders to suitable servies, without the need for ontinuous planning updates. Suh a DSS is benefiial, as it is easy to implement in the urrent pratie of ontainer transportation. Earlier proposed entralised methods an find the optimal solution for the intermodal inland transportation problem in retrospet, but are not suitable when information beomes gradually available. The main ontributions are threefold: firstly, a strutured method for reating deision trees from optimal solutions is proposed. Seondly, an innovative method is used for obtaining multiple equivalent optimal solutions to prevent overfitting of the deision tree. And finally, a strutured analysis of three error types is presented for assessing the quality of an obtained tree. A ase study illustrates the method s purpose by omparing the quality of the resulting plan with alternative methods. Keywords: Intermodal planning, synhromodal planning, ontainer transportation, deision support, deision trees. Corresponding author Bart van Riessen T: (+31) vanriessen@ese.eur.nl Eonometri Institute Erasmus Shool of Eonomis Erasmus University Rotterdam P.O. Box DR Rotterdam The Netherlands 2

3 1. Introdution Continuous growth of global ontainer volumes puts inreasing pressure on the inland road, water and rail onnetions, espeially in developed ountries with limited publi support for infrastrutural expansion. Simultaneously, shippers require more reliable inland onnetions beause their supply hain demands for just-in-time delivery, and the environmental impat of the inland transportation is inreasingly bound by restritions from governments and from shippers themselves. In this study, we onsider deision support for the planning of inland transportation. The problem is based on the situation of European Gateway Servies (EGS), an inland transportation network providing transportation between the deepsea ports in Rotterdam and various extended gates in the European hinterland (Veenstra et al., 2012, Van Riessen et al., 2014-b). Although the inland network has suffiient apaity in general, temporary ongestion ours frequently on all inland modes: road, water, and rail onnetions. Most inland transportation of ontainers is arried out by operators that are dediated to speifi modes. In the light of these developments, an integral approah for the routing and planning of all inland ontainer transportation is vital. In this study we propose a real-time DSS for providing improved planning support. In partiular, we propose to use deision trees as method in the DSS. Deision trees are a way to represent rules underlying data with hierarhial, sequential strutures that reursively partition the data (Murthy, 2005). In our method, the routing deision is made per ontainer, by applying the tree to the properties of the ontainer transportation order (i.e. booking). Figure 1 gives an example of a deision tree supporting the routing deision for a ontainer. A not A B not B Truk Figure 1 Example deision tree for deiding on the mode of transportation for a ontainer. The first deision is based on whether or not the ontainer has property A. If not, a seond deision follows, based on whether or not the ontainer has property B. Booking Container available Container due Booking lead time Transport lead time Overdue Container delivered Inland transportation Figure 2 Timeline of orders and inland transportation 3

4 1.1. Charateristi intermodal deision problem For a partiular orridor (i.e., the set of available transportation options between two loations), a set of inland servies is available, haraterized by the mode of transport (barge, rail or truk), ost per ontainer, departure time, arrival time, and vehile apaity (volume and weight). Naturally, the time between departure and arrival of a servie depends on the mode s travel speed. Typially, speed and ost are high for truking and low for barge transport. Volume apaity is high for barge transportation and low for a truk. The weight apaity for both barge and truk is mostly not restritive. The mode train has intermediate levels for speed, ost and apaity, but typially has a restritive weight apaity, espeially in mountainous regions. In this setting, we onsider sheduled barge and train servies with fixed apaities, while truks an be ordered at any time without limits. Generally, in a transportation setting, orders arrive sequentially at a transportation network planning department. Eah order has several attributes, suh as the lient name, the number of ontainers for the order, the booking lead time, the transport lead time and the size and weight of eah ontainer in the order. The number of ontainers is measured in standard ontainer sizes of Twenty feet Equivalent Units (TEU). The booking lead time is the time between the arrival of the order and the availability of the ontainer; the transport lead time is the time between the availability of the ontainers and the due time at the destination (see Figure 2). The planner s goal is to transport the ontainers at the lowest possible total ost, ideally before the due time of the ontainers, but often it is allowed to deliver a little later, indiated by the overdue time. In pratie, overdue delivery an sometimes be negotiated with ustomers, in our modelling overdue delivery is allowed at a penalty ost for the network operator. The harateristis of this deision problem do not hange over time, giving rise to periodiity, e.g. weekly. Therefore, analytis on histori information an be used to find patterns and reate a deision tree (DT). Subsequently, this deision tree an be used for deision support in future periods. B Rail Diret truk Last mile (truk) A C Figure 3 Charateristi deision problem Figure 3 illustrates an example of the harateristi intermodal deision problem shematially. We onsider 1 origin, A, and 2 destinations: B and C. Both destinations an be reahed by using a truk (T) or a rail onnetion an be used (R), for transport from A to B or from A via B to C. In the ase of using rail transportation for ontainers with destination C, last mile truking from B to C is required. In general, rail is heaper than truk, but has limited apaity. Truking apaity is abundant, and onsidered unlimited for this study. The train has a limited apaity, denoted by K. The osts for the four transportation options are denoted as dm, where d denotes the destination and m denotes the used mode, d {B, C}, m {R, T}. E.g., BR denotes the osts for transporting one unit from A to destination B by rail. Typially, the goal is to maximise the utilisation rate of the lower pried rail mode for the highest yielding destination. Beause of different transportation restritions between ontainer lasses and differenes in osts, this deision problem is not straightforward. For the planner, who must make deisions instantaneously for inoming orders, the question is how many slots to reserve for eah destination, i.e., use a booking limit of K B slots for ontainers to destination B and K C slots for destination C, adhering to K B + K C = K Real-time intermodal planning problem Nowadays, in real ases (aording to our experienes with EGS) often a greedy approah or first ome, first serve (FCFS) approah is used for planning the ontainer transportation. In ase of a greedy approah, a 4

5 ontainer transportation order is assigned to the heapest feasible servie at the time of order arrival, i.e. the heapest servie with free apaity that travels within the ontainer s time restritions. In an FCFS approah, a booking is assigned to the earliest available servie. In both methods, an order is assigned instantaneously at the time of order arrival. The problem addressed in this study is to alloate an inoming order immediately to the most suitable inland servie, as part of the optimization of the entire orridor. Existing sientifi methods for real-time deision making for planning of inland ontainer transportation fous on finding heapest or shortest paths per transport order (Ziliaskopoulos and Wardell, 2000; Ayed et al., 2011). More advaned methods for solving the online problem require real-time automated data proessing and are less insightful to human planning operators (Nabais, 2013, Li et al., 2013). We propose a method for alloating orders to servies based on optimal historial plans Proposed method for real-time deision support In reent years, several studies have proposed optimisation methods for determining the optimal alloation of ontainers to all available inland transportation servies, onsidering apaity, osts, lead times and emissions. The proposed methods are suitable for solving the offline planning problem, in whih an optimal network plan is reated for a bath of transportation orders olletively. In intermodal networks, suh as the network of European Gateway Servies, the implementation of a entralised offline approah is diffiult for various reasons: Real-time deisions: The nature of the inland transport logistis requires a real-time approah, in whih a ustomer an get immediate feedbak on the seleted mode, route, and most importantly, the estimated time of arrival. Consequently, updates in the planning of inland transportation have large influenes on the subsequent prodution proesses, possibly resulting in an undesired asade of hanges in earlier determined plans. An improved solution method must support real-time planning deisions, without ontinuous planning updates. Inomplete information: The operation of transportation systems is often not entralised, but depends on multiple ooperating deision makers, e.g. a logistis servie provider and a transportation operator. The supply hain of ontainer logistis laks information integration (van der Horst and de Langen, 2008). Capaity and/or demand information for future periods is often not fully available. However, existing entralised optimisation methods depend strongly on omplete information from integrated and automated proesses, both for terminals, as for other parts of the supply hain. They lak the flexibility to deal with inomplete information. A method must therefore be able to provide deision support even with inomplete information. 5 Human-aware deision making: In relation to the previous aspet, planning operators manually gather information ad ho, suh as real-time information on apaity and delays. Delays are ommon as the workloads in ontainer terminals have a stohasti nature and are distributed unevenly in time (Murthy, 2005). For this, diret ommuniation between manual operators is essential (Douma, 2008). A transparent approah is required that allows the human operators to inlude manually obtained information in the deision proess. We onsider this the white box poperty of the desired solution method. In this paper, we propose a general method for a real-time DSS that addresses the aforementioned issues, by meeting three main requirements. Firstly, the proposed method allows real-time deision support for alloating inoming transport orders diretly to available inland servies, resulting in a stable solution and instant feedbak to the ustomer without the neessity of ontinuous planning updates. Seondly, the method does not need omplete information to provide deision support: The deision tree obtained by our method an be applied in real-time in daily pratie without an automated deision system. Thirdly, the proposed method provides deision support in a white box system. The human planner responsible for a entral network planning an hek available apaity on a proposed servie manually and inlude that in the deision proess. In a ase study, we will exemplify how the proposed method adheres to these three harateristis while resulting in a solution loser to the optimum than the urrent pratie, although some mislassifiations are

6 still to be expeted. It is assumed that the problem is repetitive over time: the weekly servie shedule and the distribution of demand are onsidered as onstant in time Approah The model we propose is based on an analysis of the solution spae of an offline optimization model, and translates the offline model s optimal solutions to a deision tree: a white box representation of deision rules. It will therefore more easily be aepted for use in daily pratie. The method must operate well under various irumstanes and therefore multiple histori order arrival patterns are used. The proposed approah is valid for a deision proess with repetitive patterns, for whih offline optima of histori data an be determined. The quality of the real-time deision support will depend on the problem struture. We introdue a method to assess that a priori (i.e. before applying it in real-time). In the remainder of this paper, we fous on the problem of intermodal hinterland transportation. In Setion 6 we will provide insight on how to apply the proposed method to other ases as well. The approah is pitured shematially in Figure 4. First, the data of histori demands are assembled, i.e. the histori ontainer transportation bookings. Seondly, the optimal transportation plan is determined using the reently proposed linear ontainer alloation problem with time restritions (LCAT) (Riessen et al., 2014-a). The resulting optimal solutions for historial demand periods provide the baseline for real-time deision support and are used in the third step to find properties of an effetive planning of a ontainer onsidering the unertainty in the demand. The relations between ontainer properties and the planned mode and route for that ontainer in the optimal solution are determined. For this, we use a method of supervised learning for reating a deision tree based the alloations in optimal plans of the training sets. The supervised learning algorithm reates deision rules for the alloation of a ontainer to a suitable servie based on the ontainer and order properties, suh as the time of availability, the transportation lead time, and ontainer mass. Subsequently, the set of rules an be used in a real-time setting as DSS: for eah inoming order the DSS will provide a human planner with a set of suitable servies in an understandable way. 1 Data assembly Histori demand patterns 2 LCAT Optimization (offline) 3 Create deision tree Post-optimality analysis of LP 4 Deision support Heuristi for real-time use Figure 4 Proposed methodology for real-time deision support based on optimal solutions obtained offline It is expeted that the performane of the proposed method is omparable to a low-level assignment strategy suh as first-ome, first serve for ases with orders that are entirely randomly distributed aross the seleted input features. If, however, histori information ontains speifi demand patterns, identifiable from the seleted features, our method will apture those patterns without further detailed analysis. In this artile we extend our previous work on real-time deision support based on offline optimisation (Van Riessen et al., 2014-) Contribution The main ontributions of the method are threefold. First, we propose a strutured method for reating deision trees from optimal solutions. Seondly, we use a method for obtaining multiple equivalent optimal solutions to prevent overfitting of the deision tree. And finally, we develop a strutured analysis of three error types for assessing the quality of an obtained tree and the expeted error ompared to the theoretial optimum. The remainder of this paper is organised as follows. Setion 2 provides an overview of relevant literature on real-time deision support and deision trees. Subsequently, Setion 3 gives a formal desription of the proposed method for obtaining and using the deision tree. In this setion, we also ompare the results of the proposed method with an optimal strategy for the harateristi intermodal problem. In Setion 4 methods for estimating the performane of the algorithm in general are desribed. In Setion 5, the method is applied in a more general ase study of an intermodal hinterland transportation orridor, i.e. in this ase, multiple modes are available for transporting ontainers between two loations. This ase study is motivated 6

7 by a pratial ase in the port of Rotterdam. Setion 6 summarises the findings of the study and provides an outlook on future researh. 2. Literature overview of real-time deision-making (using offline models) 2.1. Optimisation models for deision problems In the literature, several methods have been proposed to deal with unertainty in a ombinatorial optimization problem. Gal and Davis (1979) desribe parametri analysis in linear programming (LP); Jenkins (1990) desribes parametri analysis in (mixed) integer linear programming (MIP). His purpose is to study the sensitivity of the problem s objetive for ertain parameters. Greenburg (1998) gives an extensive overview of the more general onept of post solution analysis for both LP and MIP. He mentions the onept of stability analysis: finding the set of parameters for whih a given solution remains optimal. However, Wallae (1996) illustrates that sensitivity analysis is only appropriate for deterministi problems and not suitable to support deision making under unertainty. An alternative for parametri analysis of a solution is to develop a robust planning, that is optimal onsidering the unertainty of all parameters. Sine 2000, a large number of studies have been published on Robust Optimization, see Bertsimas et al. (2011) for an overview. Both parametri analysis and robust optimization assume a fixed set of deision variables. However, in our ase, the number of deisions to take depends on the number of transportation orders. Branley et al. (1997) desribe the onept of post-evaluation analysis. The purpose of suh an analysis is to desribe the deision surfae, representing the value of the objetive of the problem for a set of deision variables. Suh a surfae gives insight in the effet of ertain deisions, but reating suh a surfae is omputationally very demanding as all optimal solutions must be found. This problem belongs to the lass of #P-omplete problems and is as least as diffiult as NP-omplete problems (Valiant, 1979). In post-optimization analysis, only a set of optimal solutions is studied, i.e., a subset of solutions with equal optimal objetive value (Venkat et al., 2003). Suh postoptimization analysis is often applied in multi-objetive deision-making. The intermodal deision problem an be onsidered as a speifi type of multi-knapsak problem. The multiknapsak problem is a well-known problem in literature, e.g. Rinnooy Kan et al. (1993), Pak and Dekker (2004) and Van Hentenryk and Bent (2009). The former two do not address deision support for real-time deisions. Van Hentenryk and Bent (2009) address a ategory of problems for whih the unertainty does not depend on the deision-making. Typially, the time-ritial nature of the deisions in suh a problem require online deision making. Online antiipatory algorithms (OAA) are used to solve this type of problems, by ombining online algorithms and optimisation models. For making a deision at a given point in time a distribution of future events is assumed. If a preditive model is not available, sampling of histori data an be used. With OAA, an optimal poliy that presribes the required ation in any state is not neessary, instead, only the deision provided the urrent situation and the expeted future events are onsidered using optimization models. An OAA an be used in an iterative manner, alternating between deision-making and inorporating new observations. However, in our proposed method we do not yet onsider iterative learning. The distributions of inputs are sampled and learned independently from histori data of the underlying deision proess (Van Hentenryk and Bent, 2009, Van Hentenryk et al., 2010). Verwer et al. (2014) propose a method that an be seen as the inverse from our proposal: Their tehnique enodes optimisation models from learned deision trees, whih an be used in aution settings. The method we propose uses a new approah for postoptimization analysis. As no preditive model is available, generally, we propose a learning algorithm that translates histori optimal solutions of the offline problem into real-time deision support Deision Support Systems in ontainer transportation Steiger (1998) states that the purpose of a deision support system is to provide understanding to the deision maker. Apart from the solution to a model, this also requires insight in the model and model outome. Giboney (2015) shows that the representation of a knowledge systems is ruial for user aeptane. In relation to ontainer logistis, existing literature mostly fouses on modelling and finding solutions. Several DSSs have been developed that fous on the operational problem of ontainer transhipment in ontainer terminals (e.g. Murty et al., 2005; Ngai et al., 2007; Ursavas, 2014). Ursavas (2014) also mentions some works 7

8 in literature that have proposed DSSs for ontainer transportation problems. Some studies onsider that demand is known in advane, suh as Shen and Khoong (1995), who developed a mathematial model for planning the distribution of empty ontainers, suggesting speifi options to the deision maker. Bandeira et al. (2009) propose a omputer-based heuristi for solving a network flow model of both empty and full ontainers. After eah time step, newly arrived future demand is onsidered. Janssen et al. (2004) desribe an automated planning program for ontainer truking. Several studies onsidered a stohasti model using the expeted future demand, e.g. Cheung and Chen (1998). However, beause of the problem omplexity and onsequential omputation time, this type of problems is impratial for appliation in real-time, as well as that no theoretial distribution of the future demand is available. More advaned methods for solving the online problem inlude Nabais (2013), who uses model preditive ontrol to ahieve a required modal split, and Li et al. (2013), who developed a sequential linear programming approah. All methods use omputerised systems for providing proposed solutions to the deision maker. Table 1 provides an overview of mentioned literature for deision support systems in real-time ontainer transportation planning. For eah of them is indiated to what extent they meet the 3 requirements for our problem as introdued in Setion 1.3. From Table 1 we an see that all methods support deisions in real time to various extent. Several an deal with inomplete demand information, e.g. by using distributions of future demand. Most tehniques require omplete information on apaity, exept for Ziliaskopoulos and Wardell s method, in whih apaity is not onsidered. Also, none of the existing methods have a white box representation of the deision proess. Therefore, our approah is ompared with two simple heuristis that do meet all 3 requirements: a greedy approah and a FCFS approah. Table 1 Overview of deision support tehniques in literature with respet to 3 requirements Tehnique Real-time deision Level of information required White support Demand Capaity box Shen and Khoong (1995), Network flow Periodially Complete Complete N Cheung and Chen (1998), Stoh. Progr. Time restritive* Distribution Complete N Ziliaskop. and Wardell (2000), Shortest p. Instantly Current order Ignored N Janssen et al (2004), Multi-step heuristi Periodially (15m) Ative orders Complete N Bandeira et al (2009), Network flow Periodially Distribution Complete N Nabais (2013), Model preditive ontrol Instantly Distribution Complete N Li et al (2013), Sequential LP Periodially Complete Complete N Cormen et al. (1990), Greedy Instantly Current order On request+ Y Ishfaq and Sox (2012), First Come First Serve Instantly Current order On request+ Y Proposed method, Deision Tree Instantly Histori orders Histori+ Y * The proposed stohasti programming approah an be applied periodially, but may take very long to solve + With Greedy and FCFS the operator an request up-to-date apaity info per inoming order; this is also possible in our proposed method 2.3. Deision trees We selet deision trees as the lassifiation approah for our study for several reasons. Firstly, deision trees provide diret insight into whih rules and riteria lead to a deision. This is important for pratial aeptane by the manual planning operators and is defined as a white box property. Seondly, deision trees an be trained using offline data. Subsequently, a deision tree an be used to distinguish between more than two lasses, i.e., different inland servies. Finally, the learning method must be suitable for using input parameters with ategorial data, for instane, the ustomer type (Kotsiantis, 2007). Huysmans et al. (2011) desribe several rule-base deision support methods: deision trees, deision tables and textual rule set desriptions. They performed empirial tests of users using these methods on various problems. For lassifiation type problems, as in our ase, deision tables result in slightly faster and more aurate results. However, as no diret methods for obtaining deision tables exist, deision trees are a suitable alternative. 8

9 The hallenge is to reate an aurate lassifier. The auray only be determined after the learning proess, i.e. by splitting the data in a training and test set, or by ross-validation tehniques (Kotsiantis, 2007). In our method we use a test set and a training set to validate the performane of the lassifier. Deision tree lassifiers are used as a method to struture omplex deision-making. The deision is split up in multiple stages of simpler sub deisions. A deision tree an be represented by an ayli direted graph, where a deision rule is assoiated with eah node (Safavian et al, 1991). To make a deision using a deision tree, the sub deisions per node are applied reursively to the parameters of the ase. The deision rule at eah node level defines what the next node of the deision proess will be. This is alled the hild-node. Nodes without hildren are alled leaf nodes and are assoiated with the final deision outomes of the tree. The generalization error of a tree is defined as the mislassifiation rate over the input distribution (Rokah et al., 2005). Typially, the goal for reating a deision tree is to find an optimal tree that minimizes the generalization error. Finding an optimal tree is an NP hard task, whih is only feasible for small problem sizes (Rokah et al., 2005). The topology of a deision tree and the deision rule at eah node an be estimated empirially, using real-world data for whih the intended outome is known, i.e., supervised learning (Arentze and Timmermans, 2004). Estimating a deision tree involves three aspets: design of the tree struture, the inferene method deision rule at eah node, and the seletion of the feature set ontaining the input parameters (Safavian et al, 1991). The tree struture and deision rules are determined using a learning heuristi, or inferene method. Rokah et al. (2005) provide an overview of inferene methods: Most often a top-down heuristi is used as inferene method, although bottom-up inferene methods also exist. The inferene of the tree usually involves a growth phase followed by a pruning phase. In the growth phase, branhes are added starting from the root of the tree while onsidering a splitting riterion. In the pruning phase, branh nodes are turned into leave nodes and the leaf nodes of that node are removed (Rokah et al., 2005). Aording to Arentze and Timmermans (2004), the following are the most widely used learning heuristis: C4.5 (Quinlan, 1993), CART (Breiman, 1984) and CHAID (Kass, 1980). All methods onsider a ondition with a single input variable as splitting riterion and use a top down indution method. Chandra and Varghese (2009) mention two popular splitting riteria: the Gain ratio (Quinlan, 1993) and the Gini index (Breiman, 1984). Lastly, the set of input parameters must be determined. Often, this is arried out in a greedy way, by adding the input parameter that adds the most value to the lassifiation auray iteratively, until no more improvement is made (Rokah et al., 2005). The indution of a deision tree via a learning heuristi requires a training set for the learning proess and a test set to evaluate the quality of the indued tree in a ross-validation. If some observations are more important than others, it is possible to add observation weights to eah observation. 3. Deision trees for real-time deision making in intermodal planning 3.1. Approah This study fouses on developing a method that is suitable for operational usage in a real-time setting. The quality of the real-time method is ompared to the quality of a theoretially optimal solution, obtained offline. In the method we propose in this study, we do not aim to formulate the online deision problem expliitly. Instead we aim to translate the results of an optimal model into rules for online appliation automatially. We do not pursue rule inferene by interviewing operational planning experts, as it has two disadvantages. This approah typially results in a few rules per man day (Quinlan, 1986). An expert system that an provide deision support for ontainer transportation planning may require a large amount of rules. As a result, the rule inferene for an entire network may be time-onsuming. Seondly, the quality of the transportation by the operational planning experts is unsure. For these reasons, we use a mahine learning tehnique to infer the deision rules based on optimal planning solutions of the offline problem. The DSS is indued in a series of steps. The approah depited in Figure 1 is formalised in Algorithm 1. Setion 3.2 to 3.4 desribe the steps 2 to 4 of Algorithm 1, respetively. 9

10 Algorithm 1 Obtain real-time deision support 1. Assemble N demand sets for training 2. Determine P optimal solutions for eah demand set using the CLCAT model. 3. Infer deision tree T based on N P solutions 4. Use deision tree T in a real-time setting on a per-ontainer planning heuristi Finding P optimal solutions using the CLCAT model The model we use to determine the optimal solution for the transportation planning is based on the earlier proposed LCAT model (Van Riessen et al., 2014-a). That model delivers optimal solutions for the planning of an entire network. For the harateristi intermodal problem studied in this artile, we introdue the simplified formulation of minimising osts on a single orridor, the Corridor Linear Container Alloation model with Time restritions (CLCAT). The set of all argo types that must be planned is denoted by demand set C and the set ontaining all servies by S. The total number of TEU of argo type that must be transported is denoted as the demand d, this demand must be transported on one of the intermodal servies or by diret truk. The set of servies inludes all available train and barge servies, but not truks, as truks do not depart at predefined servie shedules. s Eah servie has a slot apaity u s and a weight apaity of m s, departs at time T D and arrives at time T s A. Let x s denote the number of TEU of argo type C that is assigned to servie s S. Eah ontainer of lass has a weight of W and must be transported in the time window from t available to t due. The number of days that these ontainers are late is denoted by τ s. The number of TEU of argo lass assigned to a diret truk is denoted by v. No apaity or time restritions are onsidered for truking, as it is only used in exeptional ases. It is assumed that required truking apaity is readily available. The transit osts of transporting one TEU on servie s are denoted as s and the ost for diret truking of a ontainer of argo lass is denoted as t. The objetive of CLCAT is to minimise the total transportation osts of all ontainers, onsidering a penalty for overdue delivery of τ per day: min J CLCAT = ( s x s τ τ s + t v ), (1) s S C subjet to: v + x s = d C (2) s 10 x s u s s S (3) W x s m s s S (4) x s T s D x s t available C, s S (5) x s q (T s A t due ) τ s C, s S (6) x s, τ s, v 0 C, s S, (7) where the maximum apaity of servie s is denoted by u s (TEU apaity) and m s (mass apaity). Constraints (2) ensure that all demand is met. By onstraints (3) and (4), the total number of TEU on eah servie is restrited to the available apaity. Constraints (5) and (6) are the time onstraints, where onstraints (6) are for on-time delivery: τ s measures the total number of days that ontainers of argo lass on servie s are late. Finally, onstraints (7) are the nonnegativity onstraints for the three sets of variables.

11 For most problems, multiple equivalent optimal solutions of (1) (7) exist, as eah solution onsists of speifi assignments of argo to a servie. Often, some of these flows are fully exhangeable, e.g. beause multiple servies s are available with equal osts. In order to prevent overfitting on one optimal solution and to get deision rules that resemble all available optimal solutions as losely as possible, we need to determine a set of optimal solutions for demand set C. Finding all optimal solutions is a very diffiult task (Valiant, 1979), so we use a new, innovative approah, aimed at finding a random subset of the set of optimal solutions. In order to generate suh a random subset of optimal solutions, we solve the following problem P times: min rx (8) subjet to: (2) (7) (9) s x s τ τ s + t v = J CLCAT (10) s S C where x = [x s, τ s, v ] T, a vetor of all deision variables, and r is a row vetor of the same length of random numbers from the standard uniform distribution (ranging from 0 to 1). By (10), all feasible solutions of (8) (10) are optimal solutions of (1) (7). The random vetor r is introdued to get random subset of solutions out of the set of all optimal solutions, i.e., solving (8) (10) P times orresponds to generating P random optimal solutions from the set of optimal solutions to (1) (7). By definition, all these solutions have equal objetive value J CLCAT. As mentioned in the introdution, a entralised offline optimisation method suh as the CLCAT model is not useful in many pratial situations for three reasons: it does not support real-time deisions; it would require extensive IT implementation to obtain all data form the transportation system; it requires omplete information. In pratie, real-time deisions require a heuristi provides deision support per inoming order antiipating unknown information, suh as future demand. Current pratial heuristis do not do that, but only provide solutions for available information. Below, we desribe how the proposed DSS framework uses the optimal solutions on histori data in a learning algorithm to obtain a real-time DSS Deision tree inferene In order to obtain a deision tree that performs well under various irumstanes and demand sets we use N demand sets for training. Historial data provides the distribution of attribute values; the sets an be demand sets from the last N weeks for instane. It is assumed that future demand will show a similar struture as the histori demand sets. Per demand set, P optimal solutions are determined using (8) (10). In this study, we use P = 50 optimal solutions. In total, we use P plans for N demand sets, i.e. PN optimal plans. Eah observation in this set of solution denotes the alloation of x s ontainers of argo lass to servie s. Eah plan an ontain C S observations, resulting in m = NP C S observations in total for the inferene proess. For reating the tree, the importane of eah observation is proportional to the number of ontainers x s that is assigned, hene x s is used as the observation weight. We aim to reate a deision tree that predits the alloated servie, based on seleted input features from the transportation booking. The inferene of the deision tree is arried out using the CART method (Breiman et al, 1984), with Gini s split riterion: This method is suitable for disrete and nominal input features, aims for splitting on the most distinguishing feature and is also suitable for small data sets. The CART method uses reursive partitioning and selets in eah node the input feature that gives the least impurity of the hild nodes, aording to the Gini s index. The Gini index is denoted by (Rokah, 2005): G(y, Z) = 1 σ y= j Z 2 Z 2, j dom(y) where y represents the target attribute, Z represents the number of observations in the set and σ y=j Z the number of observations in the set with target value y = j. Hene, a pure node with just one lass has only observations with y = 1 and Gini index 0; otherwise the Gini index is positive. So the Gini index is a measure of node impurity. The reursive partitioning of nodes ontinues until in eah node the stopping 11

12 riterion has been reahed; we hoose to stop splitting a node that orresponds to less than 20% of the average alloation to a servie, i.e. 20% of the number of observations m divided by the number of servies s. Hene, the minimum node size is ε = 0.2 m. Further detailing the alloation would likely result in overfitting. s With eah leaf node, a table is assoiated ontaining the lass distribution in that leaf node for all observations in the training sets. If a leaf node is pure and the Gini-index equals zero, the lassifiation table for that node has only one entry: all observations in the training set assoiated with that node were assigned to a single inland servie. For leaf nodes with some impurity, the largest lass indiates the label of that leaf node. The obtained deision tree lassifies inoming transport orders by reursively making sub deisions until a leaf node is reahed. A lassifiation table is assoiation with eah leaf node, indiating the distribution of inland servies used for bookings that end in that leaf node. A human planning operator an use this list for determining the atual alloation, while he onsiders the remaining apaity and other pratial onsiderations. In the approah we desribe in this paper, we use a heuristi for the atual alloation, alled the DT heuristi in the remainder of the paper Applying the deision tree in a per-ontainer planning heuristi We introdue the DT heuristi to use the inferred deision tree for assigning inoming orders to inland servies. It an be used on a per-ontainer basis. For eah order, the splitting riterion in subsequent nodes are applied. Orders for transporting multiple ontainers an be onsidered as one, without loss of generality as the deision tree gives the same results for eah ontainer. In pratie, this proess an be arried out by a human operator using the deision tree. In order to demonstrate our approah in a systemati way we use an automated heuristi to generate the transportation plan for our ase study. Using the deision tree, an inoming transportation order is assoiated with one of the leave nodes. The DT heuristi will use the lassifiation table of that node ontaining the distribution of servies assoiated with that leaf node in dereasing order. The DT heuristi is provided in Algorithm 2, onsidering a previously obtained deision tree T. Note that the greedy strategy of step 4 in Algorithm 2 will selet a diret truk if none of the available inland servies has apaity left. The alloation proess is repeated at the arrival of every transportation order. To larify the proess of the deision tree inferene and appliation, the next setion shows an explanatory example for the harateristi intermodal problem. Algorithm 2 Deision tree heuristi using deision tree T 1. Consider inoming ontainer transportation order 2. Apply deision tree T to obtain the lassifiation table of the resulting leave node 3. Assign ontainers to the servies that have apaity left proportionally to distribution in the lassifiation table 4. If none of the indiated servies has apaity available, the ontainer is assigned to another servie aording to a greedy strategy, i.e., the feasible servie with minimum ost is seleted Complexity of pre-proessing steps and real-time operations For real-time deision support, the running time of any DSS is ritial. Our method supports real-time deisions by using a pre-proessing step based on histori data. Here we show that the omplexity of eah step is polynomial, and that the pre-proessing steps result in a signifiant redution of omplexity for the real-time deision proess. Without pre-proessing, we ould apply LP (1) (7) to find the optimal plan. The average ase omplexity of solving LP (1) (7) is determined using the result of Borgwardt (1982) for the average running time of the simplex method: O (n 3 pn 1), 1 (11) in whih n denotes the number of olumns, i.e. the number of deision variables, and p the number of rows, i.e. the number of onstraints. In our ase, the number of deision variables n = 2( C S ) + C, with C and 12

13 S denoting the number of argo types and servies, respetively. The number of onstraints p = 3 C S + C + 2 S. Substituting n and p in (11), and observing that n 1 2, the average ase omplexity of LP (1) (7) is O(( C S ) 3+α ), (12) with 0 < α 1. 2 In the first pre-proessing step, we onsider LP (8) (10), whih also has average ase omplexity (11) as only 1 onstraint is added in omparison to LP (1) (7). We solve this LP PN times (Setion 3.2), i.e. for N histori periods and P repetition per period. The seond pre-proessing step is the learning algorithm for the DT (Setion 3.3). The omplexity of the learning algorithm an be determined using the rationale of Su and Zhang (2006). In eah node, ontaining a subset of Z observations, one of l attributes must be seleted as splitting riterion by onsidering eah possible split for eah andidate attribute. This operation has omplexity of O( Z l). For eah level in the tree, the union of the subsets omprises all m observations. Hene, the omplexity for eah sublevel is of O(ml), with m = NP C S. Considering the minimum node size ε = 0.2 m, the maximum depth of the tree (the S maximum number of levels) equals m = 5 S. During the inferene proess, a maximum of 5 S levels must be ε onsidered. Hene, the omplexity of the entire inferene operation is: O( C S 2 l). (13) From (12) and (13) we an see that the average running time of the pre-proessing steps to obtain the tree is polynomial in the number of argo types and the number of servies onsidered in the problem. The algorithm an also be applied to larger sale problems. Finally, we onsider the problem of the real time deision problem as well (Setion 3.4). The deision tree is used for eah inoming order. The omplexity of applying the tree only depends on the number of levels in the tree, so it has a omplexity: O( S ). (14) Comparing (14) with (12), we see that the making deisions with the deision tree is muh less omputationally omplex than solving the LP for the optimal solution. This supports our aim to derive a deision support system that an be easily applied in real-time in pratie, leveraging information of histori optima Validation of the DT heuristi using the harateristi intermodal deision problem In this setion, we make a omparison between the optimal strategy and using our DT heuristi in the harateristi intermodal problem. For this problem, as depited in Figure 3, the question is how many slots to reserve for eah destination. It is possible to determine an optimal strategy analytially; but even for suh a simple problem it requires detailed analysis. Here we show that our DT heuristi is muh more onvenient to apply and results in results very lose to the optimum. The purpose here is not to study a very realisti ase, but merely a small example for whih a omparison with the optimum is possible. In Setion 4 we present a method for assessing the quality of the DT heuristi in more general ases, for whih omparing to the optimum is not possible. This is also demonstrated in a more elaborate ase study in Setion 5. In the harateristi intermodal deision problem, a planner has to deide between a lower pried, fixed apaity mode and a higher pried ample apaity mode. Two order types ompete for the lower pried mode, in this ase orders for two destinations, B and C. Containers for both destinations an go by a fixed apaity on rail, or using ample apaity of truking, so we onsider 4 transportation options: rail to B (BR), rail to C (BT), truking to B (CR) and truking to C (CT). For an optimal strategy, we need to determine the booking limits for rail slots toward B (K B ) and toward C (K C ), adhering to K B + K C = K, resulting in the lowest expeted total ost. A detailed overview of determining these booking limits is presented in Appendix A, here we provide the main results. 13

14 Optimal slot reservation We assume independent distributions for the number of ontainers for destination B and C during a planning period, denoted by N B and N C, respetively. For eah destination d and mode m, we onsider the number of used slots (i.e. planned ontainers), denoted by S dm with a ost denoted by dm. Note that the osts of transporting one unit from A to C via rail, CR, also inlude the osts for last mile truking between B and C. In line with the harateristi intermodal problem, we onsider the set of problems for whih the truking osts exeed rail osts for all destinations x, i.e., dt dr. The optimal booking limit on the train for eah destination an be found by minimising the total osts for all transports, denoted by J. The expeted osts are a sum of the osts for all four transportation options: E(J) = BR E(S BR ) + BT E(S BT ) + CR E(S CR ) + CT E(S CT ), (15) where E is the expetany operator. For a known booking limit K B (and K C ), we an determine the expeted utilisation on eah mode and eah destination. For this, we use a similar approah as in Van Riessen et al. (2015), in whih we used the expeted utilisation to determine an optimal produt mix. First, we determine the expeted slots used for transportation to destination B. The expeted number of train slots used for destination B is the sum of the onditional expetation if N B K B and of the ase that N B > K B : E(S BR ) = E(N B N B K B )P(N B K B ) + K B P(N B > K B ), (16) where P( ) denotes the probability of event, and E(N B N B K B ) = n P(N B = n n K B ) = n P(N B = n). (17) P(N B K B ) P(N n {1,2, } B K B ) n {1,2,,K B } By substituting (17) in (16), we get the expeted number of rail slots used for destination B: 14 E(S BR ) = np(n B = n) + K B P(N B > K B ) (18) n {1,2,,K B } The results for the other transportation options are shown in Appendix A. Substituting (4) and (A7) (A9) in (15) gives the expeted osts for a given value of K B (and K C = K K B ). The optimal booking limit for rail slots used for destination B, i.e., the optimal value of K B is then determined by solving K B = argmin E( T ). K B This is valid for any independent distributions of N B and N C. Here, we ll determine the optimal value K B for a uniform distribution of the demand to destinations B and C. E.g., for uniform distributions N B ~U(b 1, b 2 ), N C ~U( 1, 2 ), that allow a total demand exeeding the train apaity, i.e., b > K, the expeted number of rail slots used for destination B an be estimated using (18) as E(S BR ) = (K B b 1 ) b 1 + K B + 1 2(b 2 b 1 ) + K b 2 K B B b 2 b 1 E(S BR ) = K B 2 + b 2 1 2K B b 2 + b 1 K B (19). 2(b 1 b 2 ) The results for the other transportation options are presented in Appendix A. By using K C = K K B and finding the minimum of (15), we an find the analytial optimal value for K B : K B = ( 2 K )( CR CT )(b 1 b 2 ) + (b )( BR BT )( 1 2 ) (20) (b 1 b 2 )( CR CT ) + ( 1 2 )( BR BT ) The optimal value K B is the upper limit for slots that an be used for transporting ontainers to destination B by rail. Next, we will ompare the result of the proposed deision tree method with the analytial optimum Comparing DT heuristi to optimal solution for the harateristi intermodal problem Now, we will use some speifi instanes to ompare the performane of the DT heuristi with the analytial optimum for this problem. We assume a apaity of K = 40 for the train and we onsider three demand senarios. Table 2 shows the hypothetial osts and demand senarios. The ost for the more expensive mode, truking, is equal in both ases. The rail onnetion that is available travels diretly to destination B, omitting the need for a last-mile truking leg. Hene, this is the lowest pried transport. Using the rail onnetion to

15 destination C requires a last-mile truk delivery. For this transport an intermediate ost is assumed. The three demand senarios all result in a onflit between planning ontainers for destination B or C on the train. We determine the optimal booking limits (i.e. slot reservations) using (20). The resulting limits for destination B and C are denoted in Table 3. For all three instanes, we applied Algorithm 1 to obtain deision trees. As the first step, we assemble data patterns for training: we generate 20 sets of demand volumes for destination B and C using N B and N C, with a speifi arrival sequene. For the real-time appliation of the DT heuristi, the sequene of bookings is relevant. Seondly, we find the optimal solution for eah of these 20 demand sets using the CLCAT model. Thirdly, we use the results to train the deision tree on, with the following features for a ontainer transportation order j: the destination D j and the amount of ontainers with destination C that has already been ordered d C j before order i. As the deision tree inferene method is based on the demand set, we repeated the proess 3 times, reating 3 deision trees per demand senario, using independent demand sets. Figure 5 shows the resulting deision trees. If we inspet the deision trees for senario 1, we observe that the trees shown in Figure 5a and 5b result in a maximum of 3, resp. 4, train slots for destination C. This is remarkably lose to the optimum of 4. However, Figure 5 shows a tree that only uses 1 rail slot for destination C, whih is too onservative, ompared to the optimum. For demand senario 2, the deision trees shown in Figure 5d and 5e result in a maximum of 10, resp. 11, train slots for destination C, whih is lose to the optimum of 12. Figure 5f shows a very simple deision tree, that alloates all inoming orders to the train. This results in the optimum in most ases (e.g. if N B + N C 40), but in some extreme ase may allow too many slots be used for destination C (e.g. if a large number of orders for destination C arrives early). Finally, for demand senario 3, we obtain three very similar trees, in whih train slots are used for destination C (Figure 5g-i). This is again lose to the optimum of 18 slots for destination C. The results for all senarios show that the proposed method an identify the struture of the problem from the histori demand and provides a deision tree lose to the optimal solution in most ases for this harateristi intermodal problem. In the next setion, the DT heuristi will be applied to several variants of the harateristi intermodal problem with a range of parameters to validate the method for the more general situation. Table 2 - Cost matrix and three demand instanes of the harateristi deision problem Destination Truk Demand (1) Demand (2) Demand (3) B BR = 2 BT = 4 N B ~U(30,40) N B ~U(20,30) N B ~U(15,25) C CR = 3 CT = 4 N C ~U(0,20) N C ~U(0,20) N C ~U(15,25) Table 3 Optimal slot reservations for harateristi deision problem instanes Destination K B K C Demand (1) 36 4 Demand (2) Demand (3) D j = B D j = C D j = B D j = C D j = B D j = C d C j 3 d C j 4 d C j 4 d C j 5 d C j 1 d C j 2 Truk Truk Truk 15