Real-time control of freight forwarder transportation networks by integrating multimodal transport chains

European Journal of Operational Research 200 (2010) 733–746

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

Real-time control of freight forwarder transportation networks by integrating multimodal transport chains Stefan Bock * Institute of Business Computing and Operations Research, University of Wuppertal, 42097 Wuppertal, Germany

a r t i c l e

i n f o

Article history: Received 4 September 2007 Accepted 21 January 2009 Available online 7 February 2009 Keywords: Transportation Real-time control Multiple transshipment

a b s t r a c t Western European freight forwarders are continually being forced to increase the efficiency of their transportation processes because of the liberalization and deregulation of the European transport market. This paper proposes a new real-time-oriented control approach in order to expand load consolidation, reduce empty vehicle trips, and handle dynamic disturbances. This approach integrates multimodal transportation and multiple transshipments for the first time. Thus, it enables the flexible generation and adaptation of transportation processes. In order to be able to handle occurring disturbances, an optimization procedure that adapts the transportation processes is continually applied. Vehicle breakdowns or deceleration of vehicles, traffic congestion, and street blockages are integrated as possible disturbance scenarios. At the same time, dynamically incoming transportation requests are also dealt with. Moreover, cooperative agreements between freight forwarders, which are gaining increasing importance, are integrated by mapping hubs and external services. The efficiency of the new real-time approach is validated by several computational experiments. In particular, the use of the entire execution time for plan adaptation as well as the integration of multiple transshipments has shown promising results. Ó 2009 Elsevier B.V. All rights reserved.

0. Introduction Since the 1990s, the transport market in the European Union has been subject to various fundamental changes. As a major consequence of liberalization and deregulation, Western European freight forwarders are facing intense competition. This is mainly caused by the substantially lower labor costs of a large number of new competitors from Eastern Europe. As a result of a decline in freight charges, the German transport market – originally characterized by a majority of moderate-sized companies – faces a growing trend towards concentration. Specifically, larger-sized forwarders often take over small competitors in order to augment their presence in local regions. Smaller-sized forwarders counteract this by cooperating with each other in order to benefit from synergy effects (cf. Krajewska and Kopfer, 2006). Cooperation between freight forwarders usually comprises the bilateral outsourcing of partial or entire transportation processes and the collaborative construction of transportation hubs. Transportation hubs are mainly used for inefficient connections. Goods that are delivered to hubs are subsequently transported by local partners to their final destinations. In contrast to transportation planning where all decisions are based on the theoretical anticipation of future processes, a real-time-oriented control system has to ensure the feasible and efficient execution of the transportation process. Consequently, after starting the transportation processes, the control system has to react to each significant disturbance in the transportation network. Clearly, changes (e.g., traffic congestion or newly incoming transportation requests) are very likely to occur, and since they have a substantial impact on the overall efficiency of the controlled transportation processes, it is of significant importance that they are dealt with effectively. Thanks to the use of modern communication technologies, all required information is available in real-time even in a widespread transportation network (Mintsis et al., 2004; Gendreau and Potvin, 1998; Fleischmann et al., 2004b). As illustrated in Fig. 1, the current situation in the transportation network is mapped in a centralized data base. Based on this situation, a real-time control system has to adapt the transportation plan that is already in execution to the changed data. This, however, requires the integration of a sophisticated disturbance management system. * Tel.: +49 2024392442; fax: +49 2024393434. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.01.046

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S. Bock / European Journal of Operational Research 200 (2010) 733–746

Fig. 1. Information flow in a real-time control system of transportation networks.

1. Literature review Transportation planning has become a vital research area because of its practical relevance. In most literature on transportation, static approaches are distinguished from real-time (dynamic) ones. It is assumed that all parameters in static approaches are known in advance with certainty. Hence, no updates are necessary because of the absence of disturbances (Yang et al., 2004; Gendreau et al., 1999). In contrast to this, real-time concepts have to be able to handle unexpected data changes caused, for instance, by disturbances or newly incoming requests (Ghiani et al., 2003). In this paper only real-time approaches are going to be considered. Real-time approaches can basically be distinguished according to three criteria: the problem model, the degree of dynamism, and the kind of update handling (general classifications of dynamism and/or real-time approaches can be found in Lund et al. (1996), Larsen et al. (2002), Larsen (2001), Séguin et al. (1997) or Ghiani et al. (2003)). Real-time approaches are frequently defined as versions of the Vehicle Routing Problem (VRP) (Gendreau et al., 1999; Larsen, 2001; Ichoua et al., 2000; Giaglis et al., 2004; Ghiani et al., 2003; Ichoua et al., 2003; Savelsbergh and Sol, 1998 or Ichoua et al., 2006). Basing on existing real-time VRP-approaches, Giaglis et al. (2004) propose a general system architecture for urban distribution and real-time vehicle management. The real-time VRP-approach of Gendreau et al. (1999) allows continuous adaptation of the executed transportation plan. In order to attain substantial improvements, a sophisticated parallel Tabu Search procedure is applied. By making use of an adaptive memory that stores several solutions simultaneously, the approach efficiently combines intensification and diversification issues. Since the approach of Gendreau et al. (1999) is focused on LCL-freight forwarders (LCL = Less than Container Load), where weights and sizes are never limiting conditions, payload constraints are neglected. Gendreau et al. (1999) restrict immediate plan adaptation to the tour steps that are executed after arrival at the next destination. Because this may lower the adaptability of the control approach significantly, the approach of Gendreau et al. (1999) is extended in Ichoua et al. (2000). This extended approach additionally allows diversions from the current destinations if new requests are received. In order to deal with the simultaneity of plan execution and adaptation, diversions are only allowed after a predetermined period of time. However, diversions of current movements are still not possible during periods without incoming requests. Ichoua et al. (2003) extend the approach of Gendreau et al. (1999) by integrating time-varying travel times. Note that, particularly in urban transportation, reliable data about the time-dependent travel times, which vary significantly, are frequently available. Ichoua et al. (2003) show that their integration into real-time control systems can be useful. Fleischmann et al. (2004b) consider a general approach for the implementation of time-varying travel times in vehicle routing. Recently, there have been several promising vehicle routing approaches that make use of available stochastic data about the arrival of future requests. It has been shown that exploitation of stochastic information may result in considerable improvements (Bent and Van Hentenryck, 2004; Ichoua et al., 2006; Hvattum et al., 2006). Ichoua et al. (2006) extend the parallel Tabu Search of Gendreau et al. (1999) by exploiting knowledge about future request arrivals. More specifically, in order to manage the fleet of vehicles more efficiently, a so-called vehicle-waiting heuristic is proposed. It allows a vehicle to wait at a request destination if the probability of new, nearby requests is high. Similarly, Hvattum et al. (2006) propose a multistage heuristic that exploits statistical information on future customer demands. Its integration yields significant improvements. In contrast to the VRP, requests in the Pickup and Delivery Problem (PDP) do not necessarily have the depot as a pickup or delivery location. Hence, tours can comprise multiple pickups and deliveries. The VRP can therefore be interpreted as a special variant of the PDP (Savelsbergh and Sol, 1995; Nanry and Barnes, 2000; Lu and Dessouky, 2004; Pankratz, 2005). In the literature, many real-time PDP-approaches are defined as Truckload-PDP or Single-Load-PDP. In a Truckload- or Single-Load-PDP each vehicle can transport only one request at a time (Yang et al., 2004; Fleischmann et al., 2004a; Gutenschwager et al., 2004). Gutenschwager et al. (2004) consider a SLPDP with precedence constraints in order to deal with a real-world dispatching task of electric monorail load carriers. Fleischmann et al. (2004a) deal with real-time control in urban transportation. They integrate the handling of dynamically changing traffic information. In order to assign incoming requests to empty vehicles, specifically designed priority rules are applied. In contrast to this, Yang et al. (2004) consider incoming requests as the only source of dynamism. By assuming revenues to be proportional to the travel distance, they allow requests to be rejected. Stumpf’s (1998) extended PDP-approach integrates the definition of restrictions resulting from legal limits on maximum driving times. In order to take advantage of full-load rates, load consolidation is integrated into tour planning. Therefore, possible transshipment locations are mapped. By estimating the resulting costs with a so-called ring model (Fleischmann, 1998), pickup and delivery locations of each re-

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quest are determined in a preceding offline step. However, by assigning each request to one vehicle, resulting tours cannot integrate additional transshipment activities. In order to deal with dynamism, tour planning is repeated at predetermined intervals. In this approach, tours that are already in execution cannot be changed anymore. Pankratz (2005) considers the PDP with time window constraints (PDPTW). In order to provide efficient transportation plans, the problem is separated into grouping (or clustering) and routing. Grouping, i.e., the assignment of requests to vehicles or partners, is assumed to be dominant over routing and is optimized by a specifically designed genetic algorithm. Routing decisions are based on grouping and are made by an insertion heuristic. A dynamic version of this approach, which is based on a rolling time horizon and integrates outsourcing of complete requests to partners, is proposed by Pankratz and Gehring (2001). An analysis of existing real-time approaches with regard to the requirements of freight forwarders reveals that many important aspects are neglected. For instance, none of the real-time approaches integrates transshipments or collaborative activities (e.g., transportation hubs, dynamic outsourcing of partial or entire transportation services) into dynamic tour (re)planning. Moreover, sophisticated real-time approaches that adequately handle the simultaneity of plan adaptation and plan execution are only proposed for the VRP (Gendreau et al., 1999; Ichoua et al., 2000 or Ichoua et al., 2003). 2. The update handling of the new control approach In what follows, the update handling of the new real-time control approach is introduced. Note that the update handling determines when and how the current transportation plan is updated according to the dynamically changing situation in the transportation network. It is clear that possible changes are restricted to decisions that have not yet been made. However, since plan execution and plan adaptation take place simultaneously, the set of those decisions that are alterable by plan adaptation changes dynamically. As illustrated in Fig. 2, the proposed update handling works on two different plans (the relevant and the theoretical plan) stored at two different levels (the process level and the adaptation level). While the execution of the relevant plan takes place at the process level, future plan modifications are tested at the adaptation level. Consequently, the adaptation level deals with a theoretical plan. Since this plan is generated for a potential replacement of the current transportation plan (i.e., the relevant plan), it comprises only future decisions of the transportation process. There are two sets of decisions in this approach: the first one comprises decisions that can still be changed, and the second one consists of decisions that are already determined. As these two sets change continuously, replanning is done in a rolling horizon fashion; i.e., the total time lapse is separated into a sequence of short uniform time intervals, denoted as anticipation-horizons. At the beginning of each new anticipation-horizon, all decisions of the relevant plan in this time period are frozen; i.e., they become unchangeable. Thus, each new anticipation-horizon opens with a simulation of the relevant plan for this time interval. After conducting the simulation, a future problem that arises at the end of the new anticipation-horizon is generated. It defines the new temporary optimization problem being considered at the adaptation level. A solution to this problem determines the transportation paths of each request and vehicle. These paths commence at the locations where the particular vehicle/request will be at the end of the current anticipation-horizon. Additionally, all variables of the resulting problem are alterable throughout the entire anticipation-horizon; i.e., the set of variables is kept unchanged. Therefore, a static problem is examined at the adaptation level in the remaining time of each anticipation-horizon. If the theoretical plan outperforms the relevant plan, the theoretical plan is implemented instead. Since the theoretical plan solely comprises decisions that are relevant after the current anticipation-horizon, a possible displacement is checked only at the end of each anticipation-horizon. In order to maintain the practicability of the approach even in situations with extremely high disturbance rates, incoming disturbances are buffered during each anticipation-horizon. All buffered disturbances are iteratively integrated into the relevant and the theoretical plan after the expiration of each anticipation-horizon. Since these disturbances may have a significant impact on the resulting transportation costs, a possible displacement of the relevant plan is checked immediately after their insertion. The efficiency of the real-time control approach mainly depends on the quality of the applied improvement procedures that continuously adapt the transportation plan. In particular, it is of significant importance that the applied optimization techniques quickly yield a high improvement rate. This is because improvements can only be implemented if their calculation is completed before the decisions are made.

Process level Relevant plan currently in execution Decisions of the relevant plan Decisions of the relevant plan within the within the anticipation−horizon [t,2t] anticipation−horizon [0,t]

0

t

Decisions of the relevant plan within the anticipation−horizon [2t,3t]

2t

3t Time

Current anticipation−horizon Fixed decisions (The theoretical plan follows the guidelines of the relevant plan)

Subsequent anticipation−horizon

Subsequent anticipation−horizon

Changeable decisions

Decisions of the theoretical plan Decisions of the theoretical plan Decisions of the theoretical plan within the within the within the anticipation−horizon [0,t] anticipation−horizon [t,2t] anticipation−horizon [2t,3t] Theoretical plan currently under modification

Adaptation level Fig. 2. Illustration of the control process using an anticipation-horizon of t time units.

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3. The mathematical model This section introduces a new dynamic model that defines temporary optimization problems being considered at the adaptation level. These problem instances are derived from particular snapshots of the executed transportation process. As depicted in Section 2, these snapshots are generated at the end of each anticipation-horizon. The dynamic model integrates multiple transshipments, the use of transportation hubs, partial or total outsourcing of transportation services, and several dynamic aspects. In order to enable multiple transshipments in dynamic tour planning, request tours are defined in addition to the ordinary vehicle tours. Note that integrating transshipment activities means that requests are no longer unambiguously assigned to one vehicle but may use several vehicles. Consequently, request tours can comprise transportation activities that are executed by vehicles of the forwarder’s own fleet (including rented vehicles), by partners, or by hubs. Note that the latter two are external services with predetermined service times and costs. Since it is assumed that the freight forwarder is engaged in cooperation, these service times and costs are already determined and should comprise all additional cooperation costs (e.g., time required to establish meetings, the adaptation of staffing levels, and training). In order to improve the legibility of the following model definition, restrictions dealing with regulations on driving times and rest periods for drivers are omitted. However, these restrictions are integrated into the tour planning of the applied solution approaches. Whenever the maximum driving time is reached, the applied tour planning procedures integrate a minimum rest period. Within the subsequent model definitions, the following abbreviations, which denote specific base units, are used: ½, i.e., without any base unit; [MU], i.e., monetary units; [TU], i.e., time units; [WU], i.e., weight units; [VU], i.e., volume units. 3.1. Parameters R V

number of transportation requests (index r with 1 6 r 6 R). [] number of vehicles that are available for transportation. Vehicles either belong to the fleet of the forwarder or are rented on a shortterm basis (index v with 1 6 v 6 V). [] number of cooperating partners who provide external transportation services on a short-term basis. For example, a partner may be another forwarder, a railway company, or an airline. Prices and guaranteed execution times of external services are negotiated in advance and are therefore predetermined (index p with 1 6 p 6 P). [] number of locations in the road network (index i (or j; k) with 1 6 i; j; k 6 N). [] number of routes in the road network (index s (or t) with 1 6 s; t 6 S). [] definition of the route s in the road network, i.e., ms;i;j ¼ 1 () route s leads from location i to location j. []

P

N S ms;i;j

earliest point in time from which the tour of vehicle v can be modified. In what follows, t AV v denotes the availability time of vehicle v. [TU] earliest point in time from which the tour of request r can be modified. If, for instance, a loading activity of request r is currently in execution, tAR r determines its completion. [TU]

tAV v t AR r V;f

V;c

R;d

R;c

V;f

V;c

lh;i

final/current location of the vehicle v, i.e., lv ;i =lv ;i ¼ 1 () location i is the final/current location of vehicle v. [] R;d R;c delivery/current location of request r, i.e., lr;i =lr;i ¼ 1 () location i is the destination/current location of request r. Because this is a real-time scenario, requests may have already been picked up. Consequently, each request tour has to be determined from the current location of the request to the delivery location. [] H location of hub h in the network, i.e., lh;i ¼ 1 () hub h is located at location i. []

t Vv ;s cVv ;s t Pp;r;i;j

expected time that vehicle v needs for route s. [TU] transportation costs incurred by vehicle v for using route s. [MU] expected time that partner p needs to transport request r from location i to location j. [TU]

cPp;r;i;j tH h;j

negotiated transportation costs charged by partner p for transporting request r from location i to location j. As indicated above, additional costs incurred by cooperation should be included in these parameters. [MU] hub service time; i.e., expected time that partners need to transport goods from hub h to location j. [TU]

cH h;j

costs charged for using hub h in order to transport goods to location j. [MU]

lv ;i =lv ;i lr;i =lr;i H

v

time duration necessary for loading/unloading request r onto/from vehicle . [TU] t Lr;v =t UL r;v RUL costs incurred for loading/unloading request r onto/from vehicle . [MU] cRL r;v =c r;v ARV ARV assignment of request r at time tAR at time tAR br;v r , i.e., br;v ¼ 1 () request r is loaded onto vehicle r : [] AVR AVR AV bv ;r load of vehicle at time tv , i.e., bv ;r ¼ 1 () request r is loaded onto vehicle at time tAV . [] v RTD tardiness costs of request r at the pickup/delivery location. [MU/TU]/[MU/TU] cRTP r =c r REP RLP RED REP ; t RLD =t r . Additionally, a late deliv½t RED r r =½t r ; t r  time windows of request r; i.e., request r must not be delivered/picked up before time t r RLP RTD RTP =t , is charged with c =c per time unit. Note that if request r was already picked ery/pickup, i.e., a delivery/pickup after time t RLD r r r r RLP up, it holds ½tREP r ; t r  ¼ ½0; 1. [[TU],[TU]]/[[TU],[TU]] weight and size of request r. [WU] and [VU] wRr =sRr V V

v

v

v

wv =sv

load capacity of vehicle

v

v in weight and size. [WU] and [VU]

3.2. Variables nRr nVv xVv ;z;s Ve t Vs v ;z =t v ;z

number of steps in the tour of request r (index w; y; z with 0 6 w; y; z 6 nRr ). [] number of steps in the tour of vehicle v (index w; y; z with 0 6 w; y; z 6 nVv ). [] binary variable defining the route used in step z of vehicle v. Specifically, xVv ;z;s ¼ 1 () vehicle v uses route s in its tour step z. [] Ve AV starting/ending time of step z in the tour of vehicle v. For simplicity reasons, we define t Vs v ;0 ¼ t v ;0 ¼ t v . [TU]

S. Bock / European Journal of Operational Research 200 (2010) 733–746

737

In order to integrate multiple transshipments and multimodal transport chains, detailed request tours are defined. Besides using vehicles of the forwarder’s own fleet, these request tours may also comprise transportation steps executed by a partner or by making use of a hub. xRr;z;i;j binary variable defining the starting and ending location of step z in the tour of request r; i.e., xRr;z;i;j ¼ 1 () request r moves from location i to location j in step z. [] Re Re AR starting/ending time of step z in the tour of request r. For simplicity reasons, we define t Rs tRs r;z =t r;z r;0 ¼ t r;0 ¼ t r : [TU] xRV r;z;v ;y

binary variable defining the vehicle used in step z of request r, i.e., xRV r;z;v ;y ¼ 1 () step z in the tour of request r is identical with step y of vehicle v. [] number of requests that are loaded onto vehicle v before tour step z of vehicle v (index q with 0 6 q 6 nRVL v ;z ). []

nRVL v ;z xRVL r;y;q;v ;z

RV determination of loading activities. More specifically, xRVL r;y;q;v ;z ¼ 1 () xr;y;v ;z ¼ 1, and it holds that request r is the qth request that is loaded onto vehicle v in advance of tour step z of vehicle v. Clearly, this variable is derivable from the tour plan. Its definition, however, simplifies the following depictions. [] binary variable defining the partner used in step z of request r, i.e., xRP r;z;p ¼ 1 () step z in the tour of request r is executed by partner p . [] binary variable defining the hub used in step z of request r, i.e., xRH r;z;h ¼ 1 () step z in the tour of request r is executed by making use of hub h. []

xRP r;z;p xRH r;z;h

In order to simplify the following definitions, we introduce additional fixed variables xRV r;z;v ;y for the (theoretical) request tour steps z ¼ 0 and z ¼ nRr þ 1, as well as for the (theoretical) vehicle tour step y ¼ nVv þ 1. Hence, ARV

RV V  xRV r;0;v ;0 ¼ br;v and xr;0;v ;z ¼ 0; 8r 2 f1; . . . ; Rg; 8v 2 f1; . . . ; Vg; 8z 2 f1; . . . ; nv þ 1g

¼ 0; 8r 2 f1; . . . ; Rg; 8v 2 f1; . . . ; Vg; 8z 2 f0; . . . ; nVv þ 1g, and  xRV r;nR þ1;v ;z r

¼ 0; 8r 2 f1; . . . ; Rg; 8v 2 f1; . . . ; Vg; 8y 2 f1; . . . ; nRr þ 1g  xRV r;y;v ;nV þ1 v

3.3. Restrictions Each step of the request and vehicle tours is unambiguously defined.

8r 2 f1; . . . ; Rg : 8z 2 f1; . . . ; nRr g :

N X N X i¼1

xRr;z;i;j ¼ 1

V

8r 2 f1; . . . ; Rg : 8z 2

f1; . . . ; nRr g

:

nv V X X

v ¼1

8v 2 f1; . . . ; Vg : 8z 2 f1; . . . ; nVv g :

ð1Þ

j¼1

xRV r;z;v ;y þ

y¼1

S X

P X

xRP r;z;p þ

p¼1

H X

xRH r;z;h ¼ 1

ð2Þ

h¼1

xVv ;z;s ¼ 1

ð3Þ

s¼1

Each vehicle tour is connected.

8v 2 f1; . . . ; Vg : 8z 2 f1; . . . ; nVv  1g :

S X S X

xVv ;z;s  xVv ;zþ1;t 

s¼1 t¼1

The tour of vehicle

v starts at its current location lV;c v ;i

8v 2 f1; . . . ; Vg :

S X

N X N X

xVv ;1;s 

s¼1

8v 2 f1; . . . ; Vg :

S X

i¼1 V

xv ;nV ;s  v

s¼1

i¼1

! ms;i;j  mt;j;k

¼1

ð4Þ

j¼1 k¼1 V;f

and ends at its final location lv ;j .

! V;c

ms;i;j  lv ;i

j¼1

N X N X i¼1

N X N X N X

¼1

ð5Þ

¼1

ð6Þ

! ms;i;j 

V;f lv ;j

j¼1

Each request tour is connected.

8r 2 f1; . . . ; Rg : 8z 2 f1; . . . ; nRr  1g :

N X N X N X i¼1

xRr;z;i;j  xRr;zþ1;j;k ¼ 1

ð7Þ

j¼1 k¼1 R;c

R;d

The tour of request r starts at its current location lr;i and ends at its destination lr;j .

8r 2 f1; . . . ; Rg :

N X N X i¼1

8r 2 f1; . . . ; Rg :

ð8Þ

j¼1

N X N X i¼1

R;c

xRr;1;i;j  lr;i ¼ 1

j¼1

R;d

xRr;nR ;i;j  lr;j ¼ 1 r

ð9Þ

Correspondence of vehicle and request tours; i.e., if request r is transported in step z by vehicle v (step y of vehicle v), the chosen route and Rs Ve Re the starting and ending times (tVs v ;y ¼ t r;z , t v ;y ¼ t r;z ) are identical in both tours.

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8r 2 f1; . . . ; Rg : 8z 2 f1; . . . ; nRr g : 8v 2 f1; . . . ; Vg : 8y 2 f1; . . . ; nVv g : ð1  xRV r;z;v ;y Þ þ xRV r;z;v ;y 

N X N X i¼1

Rs Ve Re xRr;z;i;j  ð1  t Vs v ;y þ t r;z Þ  ð1  t v ;y þ t r;z Þ 

S X

!!

xVv ;y;s  ms;i;j

¼1

ð10Þ

s¼1

j¼1

Rs Ve Re Note that, since the variables are integers, the product ð1  t Vs v ;y þ t r;z Þ  ð1  t v ;y þ t r;z Þ equals one if and only if both factors equal one; i.e., if it Rs Ve Re ¼ t and t ¼ t . holds: t Vs v ;y v ;y r;z r;z Accuracy of loading activities; i.e., each loading activity is unambiguously defined. RVL

8r 2 f1; . . . ; Rg : 8y 2

f1; . . . ; nRr g

: 8v 2 f1; . . . ; Vg : 8z 2 f1; . . . ; nv g : V

nv ;z X

  RV RV xRVL r;y;q;v ;z ¼ xr;y;v ;z  1  xr;y1;v ;z1

ð11Þ

q¼1 R

nr R X X

8v 2 f1; . . . ; Vg : 8z 2 f1; . . . ; nVv g : 8q 2 f1; . . . ; nRVL v ;z g :

xRVL r;y;q;v ;z ¼ 1

ð12Þ

r¼1 y¼1

Note that requests are loaded in a sequence of non-decreasing availability in order to minimize the waiting time of the vehicles; i.e., they are loaded in a sequence of non-decreasing t Re r;y1 -values. Correct integration of hub transportation steps. RH 8r 2 f1; . . . ; Rg : 8z 2 f1; . . . ; nRr g : 8h 2 f1; . . . ; Hg : ð1  xRH r;z;h Þ þ xr;z;h 

N X N X i¼1

! H

xRr;z;i;j  lh;i

¼1

ð13Þ

j¼1

The difference between the start and end times of each vehicle tour step is lower bounded by the expected time that is needed for using a selected route.

8v 2 f1; . . . ; Vg : 8z 2 f1; . . . ; nVv g : t Vs v ;z þ

S X

tVv ;s  xVv ;z;s 6 t Ve v ;z

ð14Þ

s¼1

Each vehicle tour step commences after the completion of all loading and unloading activities. Note that it is assumed that all unloading activities are completed before the first request is loaded onto a vehicle. The subsequent loading activities are executed according to the RVL sequence defined by the variables xRVL r;y;q;v ;z . In order to simplify the following definitions, t q;v ;z determines the total loading time up to the completion of the qth loading activity that is executed before step z of vehicle v. Thus, the parameter t RVL 0;v ;z measures the time required for unloading all requests. R

8v 2 f1; . . . ; Vg : 8z 2 f1; . . . ; nVv  1g : tRVL 0;v ;z ¼

nr R X X

RV UL xRV r;y1;v ;z1  ð1  xr;y;v ;z Þ  t r;v

r¼1 y¼1

Based on t RVL 0;v ;z , completion times of all subsequent loading activities can be derived. R

RVL 8v 2 f1; . . . ; Vg : 8z 2 f1; . . . ; nVv  1g : 8q 2 f1; . . . ; nRVL v ;z g : t q;v ;z ¼

nr R X X

RVL Re L xRVL r;y;q;v ;z  ðmaxft q1;v ;z ; t r;y1 g þ t r;v Þ

ð15Þ

r¼1 y¼1 RVL Since there are nRVL determines the total duration of all these loading activities. v ;z loading activities in the tour of vehicle v before step z, t nRVL v ;z ;v ;z Hence:

RVL 8v 2 f1; . . . ; Vg : 8z 2 f0; . . . ; nVv  1g : t Ve 6 tVs v ;z þ t nvRVL v ;zþ1 ;z ;v ;z

ð16Þ

The difference between the start and the end of each request tour step is lower bounded by the expected transportation time needed by the chosen means of conveyance. V

z 2 f1; .. .; nRr g : tRs r;z þ

8r 2 f1;. .. ; Rg : 8

nv V X X

v ¼1

S X

xRV r;z;v ;y 

y¼1

s¼1

! V

V

xv ;y;s  t v ;s þ

H X h¼1

xRH r;z;h 

N X N X

! xRr;z;i;j  t Hh;j

i¼1 j¼1

þ

P X p¼1

xRP r;z;p 

N X N X

! xRr;z;i;j  t Pp;r;i;j

6 t Re r;z

i¼1 j¼1

ð17Þ It is assumed that a vehicle has to be completely empty before a request can be loaded. Therefore, the subsequent steps of the request tour are delayed until the completion of all unloading activities. V

8r 2 f1; . . . ; Rg : 8z 2 f0; . . . ; nRr  1g : tRe r;z þ

nv V X X

v ¼1

RV RVL Rs xRV r;z;v ;y  ð1  xr;zþ1;v ;yþ1 Þ  t 0;v ;y 6 t r;zþ1

ð18Þ

y¼0

Deliveries and pickups are not allowed before the particular time window opens, i.e.,

8r 2 f1; . . . ; Rg : t Re P t RED r r;nR r

8r 2 f1; . . . ; Rg :

t Rs r;1

P

tREP r

ð19Þ ð20Þ

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Transportation plans comply with the capacity constraints (i.e., payload restrictions) of the used vehicles. R

8v 2 f1; . . . ; Vg : 8y 2 f1; . . . ; nVv g :

nr R X X

V wRr  xRV r;z;v ;y 6 wv

ð21Þ

V sRr  xRV r;z;v ;y 6 sv

ð22Þ

r¼1 z¼1 R

8v 2 f1; . . . ; Vg : 8y 2 f1; . . . ; nVv g :

nr R X X r¼1 z¼1

3.4. Objective function Each transportation plan is rated according to its variable costs, i.e., according to the sum of all costs incurred by the determined transportation activities. Thus, the objective function is defined as follows

Minimize C ¼ C V þ C Tard þ C H;P þ C L þ C UL ; with

ð23Þ

C V – Costs incurred due to the movements made by vehicles. These costs may contain variable vehicle costs as well as potential tolls that are charged for using particular routes. V

CV ¼

nv X V X S X

v ¼1

xVv ;z;s  cVv ;s

ð24Þ

z¼1 s¼1

C Tard – Tardiness costs incurred due to late deliveries and late pickups

C Tard ¼

R X

RTD maxft Re  t RLD þ r ; 0g  c r r;nR r

r¼1

R X

RLP RTP maxft Rs r;1  t r ; 0g  c r

ð25Þ

r¼1

C H;P – Transportation costs charged for the use of hubs and other external transportation services R

C H;P ¼

nr X R X H X N X N X r¼1 z¼1 h¼1 i¼1

R

R H xRH r;z;h  xr;z;i;j  c h;j þ

j¼1

nr X R X P X N X N X r¼1 z¼1 p¼1 i¼1

R P xRP r;z;p  xr;z;i;j  c p;r;i;j

ð26Þ

j¼1

C L – Costs incurred by all loading activities V

R

CL ¼

nv nr X R X V X X r¼1 z¼1

v ¼1

RV RV cRL r;v  xr;z;v ;y  ð1  xr;z1;v ;y1 Þ

ð27Þ

y¼1

C UL – Costs incurred by all unloading activities V

R

C UL ¼

nv nr X R X V X X r¼1 z¼1

v ¼1

RV RV cRUL r;v  xr;z1;v ;y1  ð1  xr;z;v ;y Þ

ð28Þ

y¼1

4. Solution approaches In this section, a description of the improvement procedure that is applied at the adaptation level is given. Each plan is defined as a set of vehicle and request tours that are stored as double linked lists. The examination of neighboring solutions requires efficient operations in order to modify existing vehicle/request tours. Thus, a specific storage management system is applied. By allocating a sufficient number of elements on a stack beforehand, memory allocation operations are performed in time Oð1Þ. Determined tour plans are based on so-called relevant locations. A location l in the road network is denoted as relevant at a given point in time t if it suits at least one of the following descriptions:  l is the current location of a vehicle/request.  l is the destination/delivery location of a vehicle/request.  l is a depot or a hub location (i.e., a possible transshipment location). By restricting the definition of tour plans to relevant locations, the computational complexity in the tour determination procedure is reduced significantly. The list of relevant locations is updated at the beginning of each anticipation-horizon. However, the simulation executed for this update requires detailed tours (cf. Section 4.4). Therefore, the current relevant plan is completed at the beginning of each anticipation-horizon by temporarily inserting the shortest paths between the relevant locations. 4.1. Determination of request tours Plans are constructed and modified by an iterative (re)insertion of requests. For each request to be inserted (subsequently denoted as request r) the following alternative transport chain structures (cf. Fig. 3) are examined and assessed on the basis of the resulting costs.

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S. Bock / European Journal of Operational Research 200 (2010) 733–746

Pickup location

Pickup location Starting depot

Direct transport chain

Destination Simple multimodal transport chain

Pickup location

Delivery depot

Destination Two-level multimodal transport chain

Pickup location Depot 1

Starting depot Hub

Hub

Destination

Pickup location

Hub

Delivery depot

Destination Simple multimodal transport chain

Depot 2 Depot 3 Hub Destination Complex multimodal transport chain

Transportation process that is executed by using one’s own vehicle, a rented vehicle, or the service of a cooperating partner Hub transportation process Fig. 3. Design of the tested transport chains.

 Direct transport chains carry a request without any transshipment from its current location to its destination. The transportation process can be performed either by a vehicle of the forwarder’s own fleet or by an external partner. In order to generate an efficient tour, several suitable, randomly drawn vehicles and partners are tested. Note that the suitability of a vehicle for a transportation step of a request corresponds to the estimated additional driving time (plus potential waiting time caused by time window restrictions) that is necessary to include this step into the current tour of the vehicle.  Simple multimodal transport chains make use of a single transshipment activity. As illustrated in Fig. 3, two alternative structures arise depending on whether the transshipment depot is selected nearest to the current request location or to the delivery location. In addition, closely located hubs are also used as potential transshipment points. By testing a predetermined number of suitable, randomly drawn vehicles and partners for the resulting transportation steps, an efficient request tour is iteratively generated.  Two-level multimodal transport chains integrate two transshipment activities altogether. The first transshipment is carried out at the depot located nearest to the current location of the request. The second transshipment, however, is executed either at the depot nearest to the delivery location or at a suitable hub. By testing a predetermined number of suitable, randomly drawn vehicles and partners for executing the resulting transportation steps for both alternative structures, an efficient implementation is iteratively generated.  Complex multimodal transport chains are the most complex but at the same time the most flexible structures for defining new request tours. The generation commences by randomly drawing the number of transshipments (a) to be executed. Subsequently, the shortest path between the current location and the delivery location of the request is separated into a þ 1 subparts of nearly equal length. By determining a closely located transshipment depots, a þ 1 transportation steps arise. Finally, the entire request tour is iteratively implemented. Specifically, a predetermined number of suitable, randomly drawn vehicles and partners are tested in order to execute the a þ 1 transportation steps. In addition, alternative tour plans with a final hub transportation step are also explored. Clearly, each transport chain structure comprises one or more transportation steps. The execution of these steps is iteratively implemented for each transport chain structure. By testing a predetermined number of suitable, randomly drawn vehicles, transportation hubs, or external services of partners, the implementation that leads to minimal costs for each resulting transportation step is sought. In order to assess an alternative, the transportation costs of each transport chain structure have to be calculated. Clearly, this calculation depends on the used means of conveyance. Costs for using transportation hubs or external services of partners are directly given by the parameters cHh;j and cPp;r;i;j . Vehicle costs, however, are determined by inserting the transportation step into different existing vehicle tours in order to find a suitable alternative. Specifically, a new pickup and a new delivery have to be integrated into a vehicle tour. By cyclically moving two pointers along the tour of a particular vehicle, suitable positions for the movement to the pickup and to the delivery location are systematically sought. The first pointer addresses the tour step of the original tour that is replaced by a direct transportation to the new pickup location. Similarly, the second pointer refers to the tour step of the original tour that is replaced by a direct transportation to the new delivery location. At the beginning of the search process, both pointers are set to an identical, randomly chosen step of the vehicle tour. Consequently, in case of identical pointers, the resulting tour leads directly from the pickup location to the delivery location of the transportation step to be inserted. Otherwise, intermediate pickups and/or deliveries are possible. If the tour steps that are addressed by the two pointers already start from the pickup and/or delivery location of the transportation step to be inserted, the loading sequence of the requests at this tour location has to be adjusted accordingly. Usually requests are loaded in a sequence of non-decreasing availability at each location of a vehicle tour (cf. definition of the variables xRVL r;y;q;v ;z ). Ties are broken by smaller request numbers. Note that the availability of request r at location l of a vehicle tour is determined by the earliest point in time when the request can be loaded. Clearly, this point in time depends on the time window at the pickup location of RLP request r (½tREP r ; t r ) and on the preceding request tour of request r to location l. However, if the tour step which is addressed by the first pointer already starts from the pickup location of the inserted transportation step of request r, an immediate delivery is additionally tested if request r is deemed urgent. This scenario occurs if the timely delivery of transportation request r may be hampered by waiting time caused by subsequent loading activities at the pickup location of the inserted transportation step of request r. If this special case occurs, an urgent delivery is tested additionally. Specifically, this alternative plan transports the newly integrated request directly to its delivery location immediately after loading it. Note that, since the interrupted loading activities at the pickup location have to be resumed afterwards, a loop occurs in the vehicle tour of the alternative plan. After inserting the transportation step of the new request tour into a vehicle tour, the transportation plan needs to be synchronized. Note that, because of transshipment activities, modifications in one vehicle tour may affect timetables of other tours that subsequently transport the inserted request. In each iteration of the applied synchronization procedure, the timetable of a single vehicle is adapted. All vehicles (and tour steps) whose timetable has to be adjusted are stored on a stack during the entire synchronization procedure. Initially, the stack comprises only the vehicle (and the first tour step) that executes the inserted (partial) transportation step. After removing the

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current entry from the top of the stack, the time assignment of the subsequent tour steps is iteratively adjusted. Additionally, loading and unloading activities are sought along the vehicle tour. RVL After identifying a loading or an unloading activity, the loading and unloading times (t RVL 0;v ;z and t q;v ;z ) are corrected according to the determined time table. Additionally, the numbers of those vehicles (and tour steps) that subsequently transport an unloaded request are pushed on the stack; i.e., these subtours have to be considered afterwards. The synchronization procedure terminates when the stack is empty or an infeasible plan has been identified. Note that the latter case results from cyclical time dependencies between vehicle and request tours. In order to identify infeasible plans, it is checked whether the synchronization procedure is about to correct a vehicle tour from an identical or earlier step for a second time. After being identified, infeasible plans are discarded. After examining all four transport chain structures for the request to be inserted, the alternative that leads to minimal costs is kept for further consideration; i.e., it is kept for possible implementation. 4.2. Generating an initial solution In order to generate new transportation plans or correct existing plans that have become infeasible, an insertion procedure is applied. It iteratively inserts (or reinserts) requests into vehicle tours by making use of the tour determination procedure described in Section 4.1. V ;c At the beginning, all vehicle tours comprise a single transportation step that leads directly from the current location of the vehicle (lv ;i ) V;f to its destination (lv ;i ). In each iteration of the construction procedure, the request having the smallest time distance to one of the partially generated vehicle tours is identified and inserted. Ties are broken by lower request numbers. The time distance corresponds to the estimated additional driving time that is necessary to include the request in the nearest vehicle tour. In order to calculate the current time t distance distr between request r and the arising transportation plan, the following formula is used:

(

t dist r

(

¼ min min d

N X

!

i

R;c lr;i ; ov ;j

)

(

; dðpdr ; ov ;j Þj1 6 j 6 lv þmin d

i¼1

N X

!

i

R;d lr;i ; ov ;j

)

; dðddr ; ov ;j Þj1 6 j 6 lv j1 6 v 6 N

v

)

:

ð29Þ

i¼1

In this calculation, ðov ;1 ; ov ;2 ; . . . ; ov ;lv Þ gives the current tour of vehicle v. Furthermore, pdr and ddr denote the depot that is located nearest to the pickup and delivery location of request r, respectively. Note that these depots are possible transshipment locations for request r. Therefore, R;c R;d either the current location and/or destination of request r (lr;i and/or lr;i ) or the corresponding depots may be integrated into the arising vehicle R;c R;d tours. Thus, the sum of the minimal, estimated distance between the locations lr;i or pdr and lr;i or ddr to the nearest vehicle tour is sought. For this purpose, each vehicle tour is iteratively examined. Note that the function d defines the shortest travel time distance in the road network (plus potential waiting time caused by possible time window restrictions at the pickup and/or delivery locations) between a vehicle tour location and a request location if a timely pickup or delivery of request r is possible. Otherwise d is given the value MAXINT (i.e., a huge number). The construction procedure is also used in order to correct infeasible plans. Applied to such a situation, requests whose current transport chain is no longer feasible are iteratively reinserted; i.e., these requests are deleted and subsequently reinserted by applying the tour determination procedure described in Section 4.1. 4.3. Multi-state improvement procedure As described in Section 2, a specifically designed improvement heuristic is continuously applied at the adaptation level. This heuristic makes use of a variable neighborhood structure; i.e., the applied neighborhood operations are dynamically adapted depending on previously explored solutions. Note that the use of variable neighborhoods was originally proposed and validated for several applications by Hansen and Mladenovic´ (2001). The applied improvement procedure works in A ¼ 4 different states altogether. Each state determines the currently applied neighborhood. In state a (1 6 a 6 4) of the examination process the current tour of up to a different requests can be modified simultaneously in a single move. Each move commences by drawing the number of requests (b) to be reinserted out of the interval ½1; a. After randomly selecting b requests, the corresponding request tours are erased from the current transportation plan p. Subsequently, the selected requests are iteratively reinserted by making use of the tour determination procedure described in Section 4.1. Consequently, direct transport chains as well as simple, two-level, or complex multimodal transport chains are explored for each of these requests. After determining each new, tentative request tour, it is checked whether an improving move is still possible; i.e., it is checked whether the total costs incurred by reinsertion are still below the costs of the current transportation plan p. If not, the search process returns to transportation plan p. In order to intensify the examination in the direct neighborhood of promising solutions, the search process always returns to state one after finding an improvement. In this initial state, in which the whole improvement process commences, only a single request tour can be modified per move. Thus, the examination is focused on slight modifications. In contrast to this however, higher-numbered states allow more substantial modifications in each move because multiple request tours are adapted simultaneously. As a consequence, new promising areas of the solution space may be reached if smaller changes cannot escape from local optima. Therefore, after performing a predefined number of unsuccessful attempts, the search process changes to the next, higher-numbered state in order to diversify the search process. Note that all generated transport chains start from the current location (tAR r ) of request r. The resulting paths are therefore not restricted by already executed transportation processes. For instance, if a request is currently loaded onto a vehicle, its assignment can be changed by subsequent transshipment activities. 4.4. Preparation of each anticipation-horizon As described in Section 2, the problem that is considered at the adaptation level is generated at the beginning of each anticipation-horizon. Note that a request is reinserted by making use of the tour determination procedure described in Section 4.1. The problem-generation comprises the following steps:

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1. Integrate the consequences of the buffered disturbances that occurred during the preceding anticipation-horizon into the relevant as well as into the theoretical transportation plan. The following disturbance scenarios are dealt with:  Breakdown of vehicles: In this case, a vehicle v cannot be used for further transportation activities. This scenario is implemented by setting tVv ;s ¼ 1 for all routes. Consequently, all request tours that are making use of vehicle v are no longer valid. These tours are therefore erased from the plan and subsequently reinserted. For simplicity reasons, it is assumed that requests that are already loaded onto this vehicle become available at the destination of the last vehicle movement.  Deceleration of vehicles due to technical reasons: In this scenario, a vehicle can still be used but with a significantly reduced performance. Vehicle travel times tVv ;s are adjusted for all routes by using a vehicle deceleration factor. In order to avoid inefficient tours with considerably increased tardiness costs, this situation is handled like a breakdown scenario; i.e., the requests previously transported by this vehicle are deleted and subsequently reinserted.  Route blockages: In this scenario, a route in the used road network is blocked. By setting tVv ;s ¼ 1, the route is no longer usable for vehicles. Hence, all existing tours are examined in order to find out whether this connection is used. If so, the particular subtour is adapted accordingly.  Traffic congestion: If traffic congestion occurs on a set of routes e S, the average travel times have to be adjusted. In this case, the vehicle S are modified. Since this may affect existing shortest paths, it is handled similarly to route blockages. travel times tVv ;s on routes s 2 e Apart from the disturbance scenarios listed above, requests that occur during the preceding anticipation-horizon are iteratively inserted into the theoretical as well as into the relevant transportation plan. Both plans are now applicable to the current situation in the network and are therefore comparable. Consequently, a possible replacement of the relevant plan is checked at this point. 2. Simulate the relevant plan up to the end of the current anticipation-horizon. This simulation results in the future problem that is subsequently considered at the adaptation level. The simulation is conducted in an event-oriented manner. Events are defined as the beginning or completion of a loading activity, as a vehicle movement, or as a state transition of a request. ARV AR R;c 3. Refresh the list of relevant locations as well as the current request states (updating the parameters t AV r , t r , lr;i , and br;v ) according to the results of the simulation previously conducted in step 2. 4. Compute all shortest paths between the updated relevant locations. This calculation makes use of a modified version of the original Dijkstra algorithm. Since edge points are stored in a binary min-heap, the worst case complexity of Oðk  ðS þ N  logðNÞÞÞ arises for k relevant locations (Cormen et al., 2001). 5. Adapt the theoretical plan to the future situation which occurs at the end of the current anticipation-horizon. Since this situation is determined solely by decisions of the relevant plan, the theoretical plan may have become infeasible. The following steps are executed for its adjustment:  Simulate the theoretical plan for the current anticipation-horizon, starting with the current situation in the network.  Analogous to the relevant plan, resulting times, positions, and states at the end of the current anticipation-horizon are generated by R;c ARV this second simulation for each request (i.e., (theoretical) values of the parameters t AR r , lr;i , and br;v are obtained).  Requests with differences between theoretical and relevant values are erased from the theoretical plan and subsequently reinserted. Note that all other request tours are already feasible since they start from the actual position of the request. The improvement procedure subsequently resumes its work at the adaptation level in order to improve the theoretical transportation plan.

5. Computational results In order to validate the efficiency of the new approach and to gain an insight into real-time control systems, alternative real-time approaches are coded in C++ and applied to various process scenarios. Three different groups of transportation processes characterized by specific settings of complexity and dynamism are controlled in real-time. 5.1. Test instances and environment In each simulation, there are two groups of requests to be handled: static requests already known at the beginning of the simulation and requests that occur during the simulation. In order to avoid distorting effects caused by increasing loads that are likely to occur at the beginning of the simulation, the number of static requests (i.e., the initial load level) is made equal to the average load level. First, the construction procedure described in Section 4.2 is used in all approaches to generate an initial plan. This plan comprises the tours of the static requests. Subsequently, the improvement procedure of Section 4.3 is used for a predetermined time period. The resulting plan becomes the first relevant plan and is therefore implemented at the process level. In all real-time simulations, chronometry is based on time units (tu) comprising 45 seconds each. All real-time simulations are interrupted either after 320 tu (i.e., four hours) or after 160 tu (i.e., two hours). Similar to the initial load level of static requests, there is a final load level of unfinished transportation requests. Since this final load level is also kept approximately equal to the average load level, distorting effects at the end of the simulation can be avoided. Unfinished transportation requests at the end of the simulation lead to two different kinds of costs, which can be distinguished as costs incurred by already completed transportation steps and costs of future transportation steps. The latter are determined by the transportation plan that is executed at the end of the simulation. In order to provide a useful comparison of different real-time approaches, the sum of both costs is used. Note that it is not sufficient to compare only the transportation costs incurred so far. This results from an undefined final state of the executed transportation process at the end of the simulation. For instance, the number of completed requests as well as the resulting positions of the vehicles and requests may vary with different real-time approaches. However, since all feasible transportation plans contain a home journey at the end of each vehicle tour, an unambiguously defined situation is given only at the end of all planned processes.

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The road network that is used in all simulations comprises 5600 locations, 5 global hubs, and 57,100 routes altogether. Depending on the group of instances, the maximum number of transshipment depots varies from 210 to 399. All tested transportation processes are randomly generated based on a general problem structure. This structure was derived from practical projects carried out in conjunction with German freight forwarders. It ensures a realistic setting concerning the relative number of incoming requests, the number of used vehicles, and the average load level. Altogether three different groups of experiments, that are characterized by the following parameter settings, are generated.  Group 1/Group 2: – Jobs: 124–185 requests, fleet size: 38–55 vehicles, 5–9 external partners. – Up to 210 possible transshipment locations (depending on each instance). – Number of instances: 15 experiments (group 1), 10 experiments (group 2). – Disturbance probability: 10% (group 1), 30% (group 2).  Group 3: – Jobs: 285–372 requests, fleet size: 38–55 vehicles, 5–9 external partners. – Up to 399 possible transshipment locations (depending on each instance). – Number of instances: 10 experiments. – Disturbance probability: 20%. Note that groups 1 and 2, which comprise instances of moderate size, mainly differ in the assumed disturbance probability. Whereas disturbances rarely occur in the first group, the disturbance probability is three times higher in the second group. In contrast to the first two groups, the third group contains processes of significantly increased complexity. At the same time, an intermediate disturbance probability is assumed. Note that all generated groups of experiments do not differ with respect to the assumed tightness of the defined time restrictions. Apart from dynamically incoming requests, the simulation randomly generates disturbance scenarios with predefined probabilities. These scenarios are generated after each interval of four time units. Up to 10 new disturbances can enter the system at this point. If a new disturbance is created, it turns out to be a route blockage or traffic congestion on a set of routes with a probability of 0.15, respectively. In order to validate the adaptability of the real-time approaches in situations of considerable dynamism, more substantial disturbances are generated with higher probability. To be more specific, vehicle breakdowns in the simulation occur with a probability of 0.25 whereas vehicle decelerations are generated with a highest probability of 0.45. Since real probabilities for disturbance scenarios are highly application-dependent and therefore very variable, it is nearly impossible to anticipate them. Hence, it is not the aim of this paper to generate and test realistic disturbance scenarios, but to analyze the performance of real-time approaches under different adverse conditions. Since vehicle breakdowns and decelerations may cause significant efficiency reductions, the chosen values provide a challenging dynamic test environment. Consequently, applied realtime approaches, that ensure an efficient process execution under these adverse conditions, are very promising for practical applications. 5.2. The evaluated real-time approaches In what follows, a brief characterization of the applied real-time approaches is provided.  Continuous Adaptation Control (CAC-1/2/4): CAC-X is the new real-time approach that is depicted in Section 2. It adapts the plan that is currently in execution by continuously applying the improvement procedure that is described in Section 4.3. The parameter X 2 f1; 2; 4g denotes the length of the anticipation-horizon given in time units. For instance, CAC-2 uses anticipation-horizons of 90 seconds.  Time Limit based Control (TLbC-1/2/4): In the TLbC-X approach, adaptation of the currently executed transportation plan is triggered solely by incoming disturbances or requests. In this case, the improvement procedure described in Section 4.3 is applied for a fixed period of time of X 2 f1; 2; 3; 4g time units, i.e., for 45, 90, 135, or 180 seconds, respectively.  CACwmt -2: This approach is similar to CAC-2 except for the examined transport chain structures. The tour determination procedures applied in CACwmt -2 do not consider complex multimodal transport chains.  TLbCwmt -4: Analogous to CACwmt -2, TLbCwmt -4 excludes complex multimodal transport chains.  Pure Rule Control (PRC600 ): In contrast to all other approaches, PRC600 does not make use of improvement procedures in order to adapt the currently executed plan. Thus, the relevant plan that is generated by the preparation process (cf. Section 4.4) is not adapted. Since PRC600 does not use the multi-state improvement procedure during the execution of the transportation process, a sophisticated initial plan is of significant importance. Consequently, the computational time available to improve the initial plan is extended to 600 seconds. Note that this initial improvement process is stopped after 60 seconds in all other approaches. The following parameter configurations are used in all of the tested approaches:  Except for complex multimodal transport chains, 20 vehicles at most are examined for the possible allocation of each partial transportation step. In case of complex multimodal transport chains, this upper bound is reduced to 10 vehicles per partial process.  In each selected vehicle tour, 20 different scenarios at most are examined. Note that the use of an extra loop is excluded from this restriction.  The maximum number of transshipment processes in complex multimodal transport chains is set to two because of the results of preliminary tests. Preliminary tests indicate that approaches which allow temporary deteriorations are significantly outperformed by local search procedures. Thus, all applied adaptation processes are designed as pure improvement procedures. This underlines the specific importance of immediate improvements in a real-time environment.

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S. Bock / European Journal of Operational Research 200 (2010) 733–746

Table 1 Measured average solution quality of the different approaches. Measured results for the three tested groups of controlled processes Control approach

CAC-4 CAC-2 CAC-1 TLbC-4 TLbC-3 TLbC-2 TLbC-1 CACwmt -2 TLbCwmt -4

First group

Second group

Third group

Percentage improvement compared to PRC600

Executed moves

Percentage improvement compared to PRC600

Executed moves

Percentage improvement compared to PRC600

Executed moves

11.5 13.05 13.03 11.54 9.25 9.8 1.85 Not tested Not tested

147,430 140,577 117,893 73,096 49,879 29,654 6257

23.15 25.08 22.75 23.37 20.69 17.36 14.0 Not tested Not tested

117,299 114,555 101,245 82,755 58,304 32,468 5271

18.67 18.55 15.76 17.14 Not tested Not tested Not tested 15.34 13.83

29,482 26,771 19,352 20,016

31,730 24,197

5.3. Analysis of the measured results Considering the measured results of the different experiment groups given in Table 1, it becomes obvious that PRC600 is clearly outperformed by more sophisticated real-time approaches. The adaptability of PRC600 is quite poor because significant plan modifications are not possible. This is particularly underlined by a direct comparison of the results attained for group 1 and 2. Note that the triplication of the disturbance probability in group 2 leads to a substantial increase of the already significant quality gap between the PRC-approach and the CAC/ TLbC-approaches. Specifically, by replacing the PRC600 -approach with CAC-2, the total average costs are reduced by 13.05% for the experiments of the first group. This difference increases up to 25.08% for the problems of the second group. Clearly, the significant dynamism in the second group substantially exceeds the adaptability of the PRC600 -approach. Moreover, the measured results allow first insights into an efficient parameter setting of the real-time approaches CAC and TLbC. In particular, the length of each anticipation-horizon in the CAC-X-approaches as well as the size of the used time limit in the TLbC-X-procedures obviously has a considerable impact on the resulting costs. In case of the CAC-approaches, longer anticipation-horizons reduce the number of inefficient preparation steps (cf. Section 4.4). Consequently, more computational time is available to improve the current transportation plan. However, longer anticipation-horizons significantly restrict the search process because there are more fixed variables. Note that a similar trade-off is known for the choice of appropriate time limits in the TLbC-X-procedures (cf. Ichoua et al., 2000). Clearly, disturbance rate and problem complexity have a considerable impact on the choice of efficient time limits. This is underlined by the measured results. While the moderate-sized transportation processes of the first two groups are most efficiently controlled by making use of an intermediate anticipation-horizon, CAC-4 yields the best results for the more complex instances of the third group. Consequently, the importance of an exhaustive exploration grows with the complexity of the controlled processes. A higher disturbance rate increases the importance of immediate decisions. This becomes obvious by examining the results of the second group. Since this group comprises more complex instances than group one, a more detailed examination is required. However, longer improvement phases applied in CAC-4 do not prevail because of the significantly increased disturbance rate in group 2. A more detailed analysis of the generated solutions shows that the loss of immediate decisions in CAC-4 compared to CAC-2 is of particular importance. In addition to this, an exhaustive exploration of the real-time approach CAC-4 may even have negative side effects on some processes of group 1 and 2. An analysis of the transportation plans generated by CAC-4 shows that capacities are used almost completely. Although this leads to efficient static solutions, these plans are vulnerable to newly incoming disturbances. Hence, in order to generate more robust plans, future research has to address the integration of a sophisticated buffer management system. Based on preceding disturbances and incoming requests, it may block vehicle capacities dynamically. Since TLbC-X-approaches execute a significantly lower number of moves during the improvement processes, positive effects caused by a more comprehensive exploration prevail for all tested instances. Consequently, the approach using the largest time limit, i.e., TLbC-4, clearly outperforms the other TLbC-versions. A direct comparison of the results attained by the best CAC-X-approaches and TLbC-4 shows that a continual application of the improvement procedures results in substantial cost reductions. Specifically, average cost reductions of between 1.69 and 2.24% are attained. Note that these differences are considerable because the TLbC-4-approach already executes a large number of moves. This large number results from the high disturbance rate in the groups 1 and 2. Thus, when applying CAC-2 instead, huge improvements cannot be expected. The exploration of complex multimodal transport chains consumes considerable computational time. Thus, it is reasonable to analyze their impact. For this purpose, the modified versions CACwmt -2 and TLbCwmt -4 have been implemented and additionally applied to the transportation processes of the third group. Although these approaches can increase the number of moves by between 18.5 and 20.8%, the solution quality decreases significantly. Thus, it can be concluded that the use of multiple transshipments allows more flexible plan adaptation if transshipment costs are not prohibitive. The examination of specific processes reveals that complex multimodal transport chains are of particular importance in order to handle incoming disturbances efficiently. For instance, if a vehicle can no longer be used for transportation activities because of a breakdown, its tour is frequently divided into partial tours and assigned to vehicles already operating nearby. Consequently, besides using the entire processing time for plan adaptation, flexibility of the neighborhood operations is identified as another crucial characteristic for an efficient real-time approach.

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6. Conclusions This paper proposes a new real-time approach for freight forwarder transportation networks. In order to enable continuous plan adaptation, a dynamic model is proposed. For the first time, this new model integrates multimodal transport chains and multiple transshipments. Furthermore, the use of transportation hubs and external services that result from cooperative agreements are also considered. In order to continuously adapt the current transportation plan during its execution and handle dynamically incoming disturbances and requests, a specifically designed improvement procedure is proposed. Depending on the obtained results, the applied neighborhood operations are dynamically restricted or extended in order to intensify or diversify the search process of the improvement procedure. The measured results show that the best CAC-X-approaches outperform the other real-time approaches. Consequently, in order to control complex transportation processes in real-time, it is promising to use the entire execution time for plan adaptation. Moreover, the integration of complex multimodal transport chains is of significant importance if transshipment costs are not prohibitive. These transport chains allow the flexible handling of incoming requests and occurring disturbances. Since the results are very promising, future research should be extended to several directions. For instance, research should focus on improving the approach proposed in this paper. As indicated above, the integration of a dynamic buffer allocation system is of significant importance to the implementation of static plans. By exploiting information about incoming requests and disturbances, artificial events that are likely to occur in the near future are generated. In order to increase plan robustness, these artificial requests and events can be inserted into the vehicle tours. Recent work addressing the exploitation of stochastic knowledge about future requests reveals significant potential for real-time planning (cf. Bent and Van Hentenryck, 2004; Ichoua et al., 2006; Hvattum et al., 2006). Additional improvements of the solution quality are attainable by the efficient parallelization of the proposed improvement process. Today, almost all companies use heterogeneous Local Area Networks connecting modern Personal Computers. In most cases, however, these powerful systems are merely used by ordinary office applications, which frequently use only a small portion of the available computational capacity. Consequently, by making use of a dynamic load balancer (cf. Bock and Rosenberg, 2000), available off-peak times of the connected computers can be used for a simultaneous exploration of different areas of the solution space. Bock et al. (2006) show that the integration of parallel search procedures into real-time control systems results in considerable improvements. In particular, it turns out that the robustness of the applied real-time approaches increases significantly. Apart from improving the proposed real-time approach, future research should also consider a broader problem perspective. As indicated above, cooperation between freight forwarders is gaining increasing importance. Consequently, the new approach integrates the use of transportation hubs and external services into tour planning. This requires the negotiation of prices for transportation services. Since these prices are determined on a long-term basis, a daily tour plan and therefore the current situation in the transportation network is ignored. Consequently, transportation requests may be assigned to vehicle tours that are currently not able to provide this transportation process efficiently. 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