A recovery model for combinational disruptions in logistics delivery: Considering the real-world participators

Int. J. Production Economics 140 (2012) 508–520

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

A recovery model for combinational disruptions in logistics delivery: Considering the real-world participators Xuping Wang a,n, Junhu Ruan a, Yan Shi a,b a b

Institute of Systems Engineering, Dalian University of Technology, Dalian 116023, China School of Industrial Engineering, Tokai University, Kumamoto 862-8652, Japan

a r t i c l e i n f o

a b s t r a c t

Article history: Received 20 June 2011 Accepted 22 June 2012 Available online 13 July 2012

The existence of uncertainties may result in various unexpected disruption events in logistics delivery, which often makes actual delivery operations deviate from intended plans. The purpose of the paper is to develop a combinational disruption recovery model for vehicle routing problem with time windows (VRPTW), trying to handle a variety and a combination of delivery disruption events. Firstly, a novel approach to measure new-adding customer disruption, which considers the real-world participators (mainly including customers, drivers and logistics providers) in VRPTW, is developed. Then the paper proposes methods of transforming various delivery disruptions into the new-adding customer disruption, and determines the optimal starting times of delivery vehicles from the depot to provide a new rescue strategy (called starting later policy) for disrupted VRPTW. Based on the above, a combinational disruption recovery model for VRPTW is constructed and nested partition method (NPM) is designed to solve the proposed model. Finally, computational results are reported and compared with those of previous works, which verifies the effectiveness of the proposed solution and draws some interesting conclusions. Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved.

Keywords: VRPTW Combinational disruptions Real-world participators Recovery model Nested partition method

1. Introduction With the rapid development of e-commerce and mobile commerce, logistics delivery activities have become increasingly important in economic development and daily life. The general process of once delivery activity is: (i) customers’ requests are asked; (ii) logistics providers schedule delivery plans; (iii) drivers (delivery staff) travel according to the planed routing and serve customers as they expect. It seems easy to complete the process, especially with the help of advanced technologies nowadays! However, in the real world there are various unexpected disruption events encountered in the delivery process such as vehicles breakdown, cargos damage, changes of customers’ requests including service time windows, delivery addresses and demand amount, and so on (Bertsimas and van Ryzin, 1991; Wang et al., 2009a; Li et al., 2009b; Berbeglia et al., 2010; Yeo and Yuan, 2012; ¨ a, ¨ 2012). These disruption events often make Uskonen and Tenhial actual delivery operations deviate from intended plans, which may bring different disturbances on the participators (such as customers, drivers and providers) in logistics delivery.

n Correspondence to: Institute of Systems Engineering, Dalian University of Technology, No. 2, Linggong Road, Ganjingzi District, Dalian 116023, PR China. Tel.: þ86 411 84706593; fax: þ 86 411 84708342. E-mail addresses: [email protected] (X. Wang), [email protected] (J. Ruan), [email protected] (Y. Shi).

Existing literatures, which we will review later in the following, have put forward effective solutions for the disrupted vehicle routing problem (VRP) with a certain disruption event, but most of the proposed models and algorithms can deal with only a certain type of uncertainty. It is not easy or impossible for each proposed solution to solve actual disrupted VRP with the reality that various disruption events (vehicles breakdown, cargos damage, and changes of customers’ service time, delivery addresses, demand amount and so on) often occur successively or even simultaneously. An example is used to illustrate the combinational disruption of vehicle routing problem in Fig. 1. Fig. 1(a) shows the original routing where three vehicles serve seven customers: vehicle 1 serving customers 1 and 2, vehicle 2 serving customers 3 and 4, vehicle 3 serving customers 5, 6 and 7. Fig.1(b) shows several possible disruption events: (i) vehicle 1 breaks down when traveling to customer 2 after serving customer 1; (ii) the delivery address of customer 4 changes when vehicle 2 is serving customer 3; (iii) customer 7 decreases the demand when vehicle 3 is in the way; and (iv) there are three new-adding customers 8, 9 and 10. These disruption events may occur successively or even simultaneously, especially when the number of customers is large. One purpose of this study is to develop a common disruption recovery model for vehicle routing problem with time windows (VRPTW) which may handle a variety and a combination of disruption events. Moreover, most existing researches for disrupted VRP focus on producing new routings with the minimum costs, ignoring the real-world participators in logistics delivery (mainly involving

0925-5273/$ - see front matter Crown Copyright & 2012 Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.07.001

X. Wang et al. / Int. J. Production Economics 140 (2012) 508–520

509

Fig. 1. Combinational disruptions in logistics delivery and some recovery alternatives. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

customers, drivers and logistics providers). For some disruption event, there may be several recovery alternatives, but different participators may prefer different alternatives. In Fig. 1(b), the pink and red dotted routings represent some recovery alternatives: (i) for customer 2 who cannot be served on time because of the breakdown of vehicle 1, the pink routing which takes less waiting time is better than the red routing, but the logistics provider may prefer the red routing which can pick up the cargos in vehicle 1 to serve customer 2; (ii) customer 4 who changed the delivery address may like the pink routing better, but the provider would like to send vehicle 2 to the new address of customer 4 after serving customer 3, which need not dispatch more vehicles; (iii) for the new-adding customers 8, 9 and 10, the provider may want to dispatch vehicle 3 to serve them if there are enough cargos in the vehicle, but the driver of vehicle 3 may complain about this because he or she is too tired now; and so on. It is important to consider the satisfaction of customers and drivers (staff) which has important effects on the long-term development of delivery firms, as well as delivery costs (Sessomboon et al., 1998; Jozefowiez et al., 2008; Wang et al., 2009b; Ding et al., 2010; de Haan et al., 2012). We try applying the thought of Disruption Management to produce solutions which are satisfactory to the above three participators, which is the second purpose of the study. To sum up, the major contributions of this paper include: (i) proposing a novel approach to measure new-adding customer disruption quantificationally, considering the real-world participators in logistics delivery; (ii) designing methods of transforming different disruption events into the new-adding customer disruption; (iii) developing a combinational disruption recovery model for VRPTW and its nested partitions method. The paper is organized as follows. Section 2 reviews some related literatures. A new approach to measure the new-adding customer disruption, which considers the real-world participators in VRPTW, is developed in Section 3. Section 4 transforms different disruption events into the new-adding customer disruption, determines vehicles’ optimal starting times from the depot, and constructs a combinational disruption recovery model for disrupted VRP. Section 5 designs the nested partitions method for the proposed recovery model. In Section 6, computational experiments are demonstrated to verify the effectiveness of the model and the algorithm. Lastly, conclusions are drawn, with recommendations in future works.

2. Literature review Vehicle routing problem with time windows (VRPTW), which was proved as a NP-Hard problem, is an abstraction of vehicle scheduling problems in real-world delivery systems. Since being proposed by Dantzig and Ramser (1959), VRP has been one of research focuses in fields of operations research and combinatorial optimization. A

variety of models and algorithms for VRP have been proposed (Burak et al., 2009), and more and more researchers are taking delivery disruption events into account. Most of them have studied the dynamic vehicle routing problem (DVRP) which focuses on two kinds of uncertainties (i.e., dynamic/stochastic service requests and dynamic travel times). Bertsimas and van Ryzin (1991) developed a model for stochastic and dynamic vehicle routing in which service demands were stochastic and Bertsimas and van Ryzin (1993) extended their study to DVRP with multiple capacitated vehicles. Swihart and Papastavrou (1999) established a stochastic and dynamic model for the pick-up and delivery problem where service requests occurred according to a Poisson process and the pickup locations of the requests were independent. Secomandi (2000) applied neuro-dynamic programming (NDP) in finding approximate solutions for the vehicle routing problem with uncertain customers’ demands, compared the performance of two NDP algorithms and generated higher quality solutions. Bent and Hentenryck (2004) considered a dynamic VRPTW with stochastic customers to present a multiple scenario approach (MSA) that continuously generated routing plans for scenarios including known and future requests. Branke et al. (2005), Hvattum et al. (2006), Cheung et al. (2008), Cortes et al. (2009), Berbeglia et al. (2010), Lorini et al. (2011) and so on, have also produced various effective solutions for DVRP. Meanwhile, some scholars have studied the VRP with vehicle disruption events. Mechanical failures, accidents, and traffic jam may hinder or disrupt planned schedules (Fleischmann et al., 2004; Li et al., 2007; Lorini et al., 2011). When a vehicle on a scheduled trip breaks down, one or more vehicles need to be rescheduled to serve the customers on that trip. Fleischmann et al. (2004) described the derivation of travel time data from modern traffic information systems and presented a general framework for the implementation of time-varying travel times in various vehicle-routing algorithms. Li et al. (2007) proposed a vehicle rescheduling problem (VRSP) with vehicle breakdown disruption, and developed a prototype decision support system (DSS) to minimize operation and delay costs while serving the customers on the disrupted trip. The thought of Disruption Management, which aims at minimizing the deviation of actual operations from intended plans with minimum costs, provides an effective idea to deal with realtime and unpredictable events (Jens et al., 2001). At present, Disruption Management has been widely applied in flight scheduling, machine scheduling, supply chain coordination, and so on (Yu and Qi, 2004). Some researchers have introduced the thought into logistics delivery. Wang et al. (2007) developed a disruption recovery model for the vehicle breakdown problem of VRPTW and proposed two rescue strategies: adding vehicles policy and neighboring rescue policy. Zhang and Tang (2007) considered the vehicle

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breakdown disruption to build a Disruption Management model and adopted a hybrid algorithm of ant colony optimization (ACO) and scatter search to determine good approximate solutions. Li et al. (2009a, 2009b) imposed a penalty value for each route change to reduce the changes of the initial vehicle routes and developed a Lagrangian relaxation based-heuristic being integrated with an insertion based-algorithm to obtain feasible solutions for the disrupted VRP with vehicle breakdown disruption. Wang et al. (2009a) analyzed the negative effects of changes of customers’ requests on original delivery plan to build a multi-objective VRPTW disruption recovery model. Wang et al. (2009b, 2011) also studied the VRP disruption recovery model with fuzzy time windows, and (Wang et al., 2010) carried out a further study on the vehicle breakdown disruption. Mu et al. (2010) described the difference between disrupted VRP and classic VRP in detail, and developed two Tabu Search algorithms to solve the VRP with vehicle breakdown disruption. Ding et al. (2010) considered the uncertainty of human behaviors to construct a Disruption Management model for delivery delay problem of VRP. Existing literatures have produced effective solutions for a certain disruption event of VRP, but the delivery practice requires a common disruption recovery model which can handle a variety and a combination of disruption events, and can produce solutions satisfactory to the real-world participators in logistics delivery. We try applying the thought of Disruption Management to develop such a model and designing its algorithm.

3. Disruption measurement on disrupted VRPTW 3.1. Original VRPTW The original VRPTW studied in this study is as follows. One depot owns K delivery vehicles with the same maximum capacity. Each customer in the set of customers N should be visited once in its requested service time window. Each vehicle should leave from and return to the depot. The target is to determine the delivery plan with the shortest total delivery distance. Notations which will be used in the following are defined in Table 1. The mathematical model of VRPTW is: min

N0 K X X

N0 X

cij xijk

ð1Þ

i,j A N0 ,k A f1,. . .,K g

ð2Þ

k ¼ 1 i ¼ 0 j ¼ 0,j a i

s:t: xijk ¼ f0,1g

Table 1 Notations for original VRPTW. Notations Meanings N N0 K Q cij tij di seri qijk xijk

A set of customers, N ¼ {1, 2,y,n} A set of the depot and customers, N0 ¼ {0}[N The total number of vehicles in the depot The maximum capacity of each vehicle The distance between node i and node j, i, j AN0 The travel time between node i and node j, i, j AN0 The demand amount of node i, d0 ¼ 0 The service time at node i The available load of vehicle k between node i and node j A binary variable: xijk ¼ 1 means vehicle k travels from node i to node j; otherwise xijk ¼ 0 uik A binary variable: uik ¼1 means node i is visited by vehicle k; otherwise uik ¼ 0 Rstai The starting service time for node i [stai, endi] The time window of node i: stai, earliest service time; endi, latest service time M A large positive number

uik ¼ f0,1g K X

iA N,kA f1,. . .,K g

ð3Þ

for each i A N

ð4Þ

uik ¼ 1

k¼1 K X N X

x0jk ¼

k¼1j¼1 N0 X

xlik ¼

l ¼ 0,l a i

K X N X

xi0k r K

ð5Þ

k¼1i¼1 N0 X

xijk ¼ uik

for each group of i

j ¼ 0,j a i

and k,i A N,k A f1,. . .,K g qijk  xijk r Q N0 X

ð6Þ

i,j A N0 ,k A f1,. . .,K g

di  uik rQ

ð7Þ

for each k A f1,. . .,K g

ð8Þ

i¼1

Rstaj Z Rstai þ seri þt ij ð1xijk Þ  M stai rRstai rendi

iAN

i,j A N0 ,i oj,k A f1,. . .,K g

ð9Þ ð10Þ

where the objective function (1) is to minimize total delivery distance; constraints (2) and (3) define the values and ranges of xijk and uik; constraint (4) ensures that each customer is served only once; constraint (5) ensures that each vehicle should leave from and return to the depot and the number of dispatched vehicles should not exceed the total number of vehicles in the depot; constraint (6) represents the vehicle which arrives at customer i should leave from customer i; constraints (7) and (8) represent any vehicle should not load more than its maximum capacity at any time; constraint (9) ensures that the starting service time of customer i is earlier than that of customer j if customers i and j are served by the same vehicle, ioj; constraint (10) satisfies the requirement of time windows. 3.2. Disruption measurement on customers, drivers and providers Disruption Management aims at minimizing the negative effects or deviations caused by unexpected events on or from the original plan, so the effects or deviations should be measured quantitatively before being taken as the minimization objectives, which is called Disruption Measurement (Qi et al., 2006; Cauvin et al., 2009; Wang et al., 2009a). In Section 4, we will transform different disruption events into the new-adding customer event, so in this section the disruption measurement on disrupted VRP focuses on measuring the new-adding customer disruption. In recovery routings, the number of customers, delivery addresses, time windows and other parameters may change, so some notations in original VRPTW are labeled as new notations correspondingly: N0-N00 , xijk-xijk0 , cij-cij0 , stai-stai0 , RstaiRstai0 , endi-endi0 , and so on. (In new delivery plans the set of nodes N00 include not only the original customers without changes but also new-adding customers.) However, there are also some notations unchanged, such as the number of vehicles K and the maximum capacity of vehicle Q. 3.2.1. Disruption measurement on customers In the transportation business, time windows are not always strictly complied and the deviation of service time from specific time window determines the customer’s satisfaction level (Tang et al., 2009; Wang et al., 2009b; Hannaneh and Taravatsadat, 2010). Customers should be served on time, which may improve their satisfaction and loyalty. According to the original optimized plan, each customer can be served in its requested time window. If customers can receive the service on time after some disruption

X. Wang et al. / Int. J. Production Economics 140 (2012) 508–520

events occur, their satisfaction levels may keep unchanged; however, if they cannot be served on time because of the disruptions, they may complain about the deviation, which likely decreases their satisfaction levels. The disturbance on customers’ service times refers to that the actual arrival time is earlier than the earliest service time or later than the latest service time. Thus, in order to keep and improve the satisfaction and loyalty of customers, the recovery plan for disruption events should to the greatest extent reduce the service time deviation from the original plan. Assuming that [stai0 , endi0 ] represents the time window of customer i and arri0 represents its actual arrival time, the service time deviation of customer i is:

l1 ðsta0i arr0i Þ þ l2 ðarr0i end0i Þ, l1 , l2 A f0,1g,

ð11Þ

where, if arri0 ostai0 , then l1 ¼ 1 and l2 ¼ 0; if arri0 4endi0 , then l2 ¼ 1 and l1 ¼0; if stai0 %arri0 %endi0 , then l1 , l2 ¼0, as Fig. 2 shows. The total service time deviation of all customers is: 0

y

N X

0

ðl1 ðsta0i arr 0i Þ þ l2 ðarr 0i endi ÞÞ,

l1 , l2 A f0,1g

511

study is to minimize the deviation of total delivery costs. Delivery costs mainly depend on total travel distance and the number of dispatched vehicles, so the deviation of total delivery costs is: 0 1 N 00 N 00 N0 N0 K X K X X X X X 0 0 s@ c x  cij xijk A ij ijk

k ¼ 1 i ¼ 0 j ¼ 0,j a i

þ

K X

k ¼ 1 i ¼ 0 j ¼ 0,j a i

C K ðv0k vk Þ,v0k ,vk A f0,1g

ð14Þ

k¼1

PK

PN00

PN00

c0ij x0ijk represents the total delivery PK PN0 PN0 distance of the recovery plan; k¼1 i¼0 j ¼ 0,j a i cij xijk reprewhere

k¼1

i¼0

j ¼ 0,j a i

sents the total delivery distance of the original plan; C K is the P fixed cost of dispatching a vehicle and Kk ¼ 1 C K ðv0k vk Þ represents the change of vehicle fixed costs, where v0k ,vk A f0,1g, if vehicle k is assigned in the original plan or in the recovery plan, then vk or v0k ¼ 1, otherwise, vk or v0k ¼ 0.

ð12Þ

i¼1

4. A recovery model for combinational delivery disruptions

where y is the penalty cost coefficient of per unit time deviation. 3.2.2. Disruption measurement on drivers The delivery routing owns two main attributes: the total distance and paths (sections) in the routing. To meet the requests of newadding customers may result in changes of the total delivery distance and driving paths, which may bring disturbances on drivers. According to interviews with real-world delivery drivers, some interesting things are found: (i) drivers prefer the routings with which they are familiar because they know the exact locations, the customers and the delivery processes; (ii) drivers may accept extra delivery tasks sometimes, but they often find some reasons to refuse the rearrangement, especially when it is frequent; (iii) customers also prefer familiar delivery staff who they have trust in. Based on these findings, the study aims at minimizing the deviation of total driving paths from the original plan to reduce the disturbance on drivers. The deviation of total driving paths is represented as follows: 0

s

0

N0 X N0 X i¼0j¼0

0

c0ij 9p0ij pij 9 þ m

0

N0 X N0 X

9p0ij pij 9,i, j A N 00 ,pij ,p0ij A f0,1g

ð13Þ

i¼0j¼0

where s is the delivery cost coefficient of per unit distance; m is the penalty cost coefficient of increasing or reducing a delivery path; pij , p0ij A f0,1g, if there is a delivery path between node i and node j in original delivery routing, pij ¼ 1 , otherwise, pij ¼ 0; if there is a delivery path between node i and node j in new delivery routing, p0ij ¼ 1, otherwise, p0ij ¼ 0. 3.2.3. Disruption measurement on providers Obviously, total delivery costs which logistics providers often care about most will be affected by the new-adding customer disruption. Decision-makers often hope that not too much cost is added when some disruption events occur (Yu and Qi, 2004; Jens et al., 2009). Thus, the third objective of the recovery plan in this

In Section 3, we measure the new-adding customer disruption from aspects of customers, drivers and logistics providers. In order to develop a combinational disruption recovery model, methods of transforming different disruption events into the new-adding customer disruption will be first analyzed and developed in this section. 4.1. Delivery disruption events transformation We try to transform different disruption events such as vehicles breakdown, cargos damage, changes of customers’ requests into the new-adding customer disruption event, which will facilitate us to develop a common and combinational disruption recovery model for VRPTW. Detailed transformation conditions and ideas are as follows: (1) Transformation of vehicle disruption events There are two main kinds of vehicle disruption conditions: vehicle breakdown and vehicle blocked. When one vehicle is damaged and its remaining delivery tasks cannot be completed by itself, the unserved customers can be regarded as new-adding customers with the same requests such as delivery addresses, time windows, demand amounts and so on, as Fig. 3 shows. In Fig. 3, when vehicle 1 travels to customer 2, it breaks down. Customers 2 and 1 cannot be served by vehicle 1, so they are transformed into new-adding customers 20 and 10 whose requests are the same with original customers 2 and 1. Note that vehicle 1 will be transformed into virtual customers (Pseudo-depot) (Wang et al., 2009a; Li et al., 2009b) in the recovery plan. When one vehicle is blocked and cannot go on in a certain period, its planed delivery routing may be unchanged if it has enough time for the remaining customers. However, the customers with narrow time windows may make the original plan infeasible, so these disrupted customers will be transformed into

Fig. 2. The service time deviation of customers from the original plan.

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X. Wang et al. / Int. J. Production Economics 140 (2012) 508–520

Fig. 3. Vehicle breakdown events to new-adding customer events.

Fig. 4. Vehicle blocked events to new-adding customer events.

Fig. 5. Changes of time windows to new-adding customer events.

new-adding customers with the same requests. As Fig. 4 shows, vehicle 1 is blocked when traveling to customer 2 and customer 2 has a narrow time window who cannot wait for the blocked time, so customer 2 will be transformed into a new customer 20 . Vehicle 1 may travel to customer 1 directly after the blocking and the new customer 20 may be served by vehicle 2. (2) Transformation of cargo disruption events The cargo disruption mainly refers to cargos damage which may be caused by bad weather, deterioration, accidents or other events. If the whole cargos in some vehicle are damaged, all the remaining customers will be transformed into new-adding customers; if only part of the cargos are damaged, the customers who can be met with the remaining goods will be served according to the original plan, other remaining customers will be looked as new-adding customers. (3) Transformation of customer disruption events Customer disruption events include changes of service time windows, delivery addresses and demand amounts, removal of customers and so on.  Changes of time windows Assuming that the service time of customer i is requested earlier, its original time window ½stai , endi  will be ½stai Dt, endi Dt where Dt is a positive number. If Dt is so small that vehicle k which is dispatched to serve customer i in the original plan can squeeze out some extra time longer than Dt by speeding up, the request will be ignored and bring no disruption to the original plan. If Dt is too large that vehicle k cannot squeeze out enough time, the request will be regarded as a disruption and customer i will be transformed into a new customer i0 with time window ½stai Dt, endi Dt.

Assuming that the service time of customer i is requested later, its time window ½stai , endi  will be ½stai þ Dt, endi þ Dt. If Dt is relatively small and vehicle k can wait for the extra time at customer i with the precondition that it will bring no negative effects on the remaining delivery, the request will be ignored and no disruption is brought to the original plan. If Dt is large and vehicle k cannot wait for customer i, the request will be regarded as a disruption and customer i will be transformed into a new customer i0 with time window ½stai þ Dt, endi þ Dt. As Fig. 5 shows, the time window of customer 1 is postponed because of some reason and vehicle 1 cannot wait for the extra time considering the service time of customer 5, so vehicle 1 can travel to customer 5 directly after serving customer 2 and customer 1 is transformed into a newadding customer 10 who may be served by vehicle 2.  Changes of delivery addresses If customers’ delivery addresses change, the original plan cannot deal with the changes which will be regarded as disruptions. Assuming that the delivery coordinate ðX i ,Y i Þ of customer iis changed into ðX 0i ,Y 0i Þ, customer i will be transformed into a new customer i0 with its delivery coordinate ðX 0i ,Y 0i Þ.  Changes of demand amounts Changes of customers’ demand amounts include demand reduction and demand increase. The demand reduction of some customer brings about no disruption on the original delivery plan. Vehicles can deliver according to the planed routing, so the demand reduction is not considered as a disruption in this study. Whether the demand increase of customer i is regarded as a disruption depends on the occurrence time t and the increase

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513

Fig. 6. Changes of demand amount to new-adding customer events.

amount Dd. If vehicle k which is dispatched to serve customer i has left from the depot at time t and no extra cargos more than Dd is loaded, the increase will be regarded as a disruption; If vehicle k has left from the depot at time t but loads backup cargos more than Dd, the increase would not be regarded as a disruption. If vehicle k has not left from the depot at t and can load more cargos than Dd, the increase brings no disruption on the original plan; If vehicle k has not left from the depot at t but cannot load more cargos than Dd, the demand increase is also looked as a disruption. After the demand increase being identified as a disruption, a new customer i0 whose demand amount is Ddwill be added with the same coordinate and time window with customer i. For example, in Fig. 6 the demand amount of customer 1 increases but vehicle 1 has not enough extra cargos after serving customer 2. However, vehicle 2 may travel to the new customer 10 if it has enough cargos after serving customer 3.  Removal of customers Customers may cancel their requests sometimes because of a certain reason, but the planed delivery routing needs no changes. When passing the removed customers, delivery vehicles just go on with no service. (4) Combinational disruptions Combinational disruptions refer to that some of above disruption events occur simultaneously on one customer or several customers. This study focuses on the combinational disruption of various customer disruption events. For one customer i with the coordinate ðX i ,Y i Þ and time window ½stai , endi , if its delivery address is changed into ðX 0i ,Y 0i Þ and service time is put off, a new customer i0 can be added with the coordinate ðX 0i ,Y 0i Þ and time window ½stai þ Dt, endi þ Dt. The combinational disruption events may occur on several customers. For example, the time window of customer i is requested earlier, from ½stai , endi  to ½stai Dt, endi Dt; the delivery address of customer j is changed, ðX j ,Y j Þ-ðX 0i ,Y 0i Þ; extra demand Dd is requested by customer m. The transformation of these disruptions is shown as in Table 2 and Fig. 7. 4.2. Determination of the optimal starting times for vehicles

Table 2 Transformation of combinational disruptions from multi-customers. Original customers

Disruption events

New customers

i j m

[stai, endi]-[stai-Dt, endi-Dt] (Xj,Yj)-(Xj0 ,Yj0 ) dm-dm þ Dd

i0 : [stai  Dt, endi  Dt] j0 : (Xj0 ,Yj0 ) m0 : Dd

Note that: After being transformed into new customers, the original customers will not be considered in new delivery plan except the customers who only have demand increase disruption.

time window penalty. Sancak and Salman (2011) presented a delayed transportation policy that allows delaying the shipment of trucks with a small truckload percentage to increase vehicle utilization while still satisfying the requirements. The above works are the basis of the determination of vehicles’ optimal starting times from the depot in this study. In this study the optimal starting time of each vehicle refers to its latest departure time from the depot, because the later departure from the depot may provide more chances to cope with new-adding customers in the recovery plan. For example, if one vehicle leaves from the depot to its first customer at time 0, it may fail to serve one new-adding customer that occurs around the depot with earlier time window (e.g., [3, 6]); however, if the vehicle does not leave before its latest departure time (e.g., time 8), it may firstly serve the unexpected customer and then go to its original first customer in its updated route. Some new notations used in the following are supplemented n o here. Nk ¼ Nk0 ,Nk1 ,N k2 ,. . .,Nkm ,Nkm þ 1 is the set of the customers served by vehicle k and the depot,k A f1,. . .,K g(Nkm þ 1 is the depot, as well as Nk0 ); Lstaki represents the latest starting service time for node N ki ; BST k represents the optimal starting time of vehicle k from the depot. A change of departure time from one customer or the depot will result in different departure times at its succeeding customers (Kok et al., 2011), so we calculate the latest starting service time for each node in an inverted order (Nagata et al., 2010). n o k For each node Nki in Nk ¼ Nk0 ,Nk1 ,N k2 ,:::,Nkm ,Nkm þ 1 , ½staki , endi  represents its time window, iA f0,1,. . .,m,m þ 1g and serki repre-

Many solutions for VRPTW have developed routes by considering the feasible departure time (Kok et al., 2011), but most existing researches on disrupted VRP assume that all the assigned vehicles leave from the central depot at time 0. Although delivery vehicles may arrive at customers early, they have to wait until the earliest service time, which will result in waiting costs. In fact, the optimal starting (departure) time of each vehicle from the depot can be determined according to its delivery tasks, which may decrease total delivery costs and provide a new rescue strategy for some disruption events. Kok et al. (2011) proposed the vehicle departure time optimization problem (VDO) with time-dependent travel times and approached the VDO as a post-processing step of solving the VRPTW, and a case study demonstrated that optimizing departure times may reduce 15% duty time on average. Nagata et al. (2010) proposed an efficient calculation method of the

sents its service time. t ki,i þ 1 is the travel time between node Nki and node N kiþ 1 . The last node in the routing of vehicle k is the depot who does not need service but should be visited before k

k

time end0 , so its latest starting service time can be assigned end0 , that is, k

Lstakm þ 1 ¼ end0

ð15Þ

For the intermediate node N ki , i A f1,. . .,mg, its latest starting k service time Lstaki equals to the smaller of endi and Lstakiþ 1 k k t i,i þ 1 ser i , but it should not be smaller than its earliest service time staki , that is, n n o o k ð16Þ Lstaki ¼ max min endi , Lstakiþ 1 t ki,i þ 1 ski , staki

514

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Fig. 7. Combinational disruption events to new-adding customer events.

Table 3 The calculation results of the latest starting service time. stai

endi

seri

0

0

200

0

1

15

25

10

2

40

50

10

3

80

90

10

0

0

200

0

Nodes

Fig. 8. An example for calculating the optimal starting time of vehicles.

which guarantees the requirement of customers’ time windows. On the basis of this, the latest starting service time of Nk0 is: n n o o k Lstak0 ¼ max min end0 ,Lstak1 t k0,1 sk0 ,stak0 ¼ n n o o k ð17Þ max min end0 ,Lstak1 t k0,1 ,0 which is the latest departure time from the depot of vehicle k, that is, the optimal starting (departure) time vehicle k, n n o o k ð18Þ BST k ¼ max min end0 , Lstak1 t k0,1 ,0 Here an example is illustrated to demonstrate the calculation of the optimal starting times of vehicles. As Fig. 8 shows, time windows and travel times are given, and the service time seri for each customer is 10. According to (15)–(17), the latest starting service time of each node can be calculated, as Table 3 shows. Thus, the optimal starting (departure) time of the vehicle is 10, meaning that the vehicle can leave from the depot at time 10. Considering the optimal starting times of vehicles can provide a new disruption rescue strategy (called starting later policy in the paper) for VRPTW, because vehicles may have not left from the depot when disruption events occur. In the above example, assuming that the vehicle leaves from the depot at its optimal starting time 10, if a new-adding customer occurs or some customer changes its demand amount in the routing during time 0 and 10, these disruption events may be dealt with because of the starting later policy.

0

0

ðl1 ðsta0i arr 0i Þ þ l2 ðarr 0i endi ÞÞ

ð19Þ

i¼1

0

f2 ¼ s

0

N0 X N0 X i¼0j¼0

0

c0ij 9p0ij pij 9 þ m

0

N0 X N0 X i¼0j¼0

9p0ij pij 9

0

0

N0 K X X

N0 X

c0ij x0ijk 

k ¼ 1 i ¼ 0 j ¼ 0,j a i

þ

K X

N0 K X X

N0 X

1 cij xijk A

k ¼ 1 i ¼ 0 j ¼ 0,j a i

C K ðv0k vk Þ

ð21Þ

k¼1

s:t: x0ijk ¼ f0,1g

i,j A N 00 ,kA f1,:::,K g

ð22Þ

u0jk ¼ f0,1g

j A N 0 ,kA f1,. . .,K g

ð23Þ

for each i A N0

ð24Þ

K X

u0ik ¼ 1

k¼1 K X N X

x00jk ¼

k¼1j¼1

K X N X

x0i0k r K

ð25Þ

k¼1i¼1

0

0

N0 X

x0lik ¼

N0 X

x0ijk ¼ u0ik

for each group of i

j ¼ 0,j a i 0

l ¼ 0,l a i

and k,i A N ,k A f1,. . .,K g q0ijk  x0ijk r Q

i,j A N00 ,k A f1,. . .,K g

ð26Þ ð27Þ

0

N0 X

0

di  u0ik rQ

for each k A f1,. . .,K g

ð28Þ

i¼1

 M,i,j A N0 ,k A f1,. . .,K g

According to the above, a combinational disruption recovery model for VRPTW is constructed as follows.   min f 1 , f 2 , f 3 N X

f 3 ¼ s@

  max minf200,2010g,0 ¼ 10   max minf25,502010g,15 ¼ 20   max minf50,903010g,40 ¼50   max minf90,2001010g,80 ¼90 200

Rsta0j Z Kstak þRsta0i þ ser0i þ t 0ij ð1x0ijk Þ

4.3. A combinational disruption recovery model for VRPTW

f1 ¼ y

0

Lstai

ð20Þ

ð29Þ

Kstak ¼ BST k ,k A f1,. . .,K g

ð30Þ

l1 , l2 A f0,1gpij ,p0ij A f0,1g,v0k ,vk A f0,1g

ð31Þ

x0vck k ¼ 1

ð32Þ

8vck A VC, k A f1,2,. . .,K g

The first objective (19) is to minimize the disturbance on customers’ service time, where if arri0 ostai0 , then l1 ¼ 1 and l2 ¼0; if arri0 4endi0 , then l2 ¼ 1 and l1 ¼0; if stai0 %arri0 %endi0 , then l1 , l2 ¼0; objective (20) aims at keeping the new delivery routing as close as that in the original plan, where pij ,p0ij A f0,1g, if

X. Wang et al. / Int. J. Production Economics 140 (2012) 508–520

there is a delivery path between node i and node j in original delivery routing, pij ¼ 1 , otherwise, pij ¼ 0; if there is a delivery path between node i and node j in new delivery routing, p0ij ¼ 1, otherwise, p0ij ¼ 0; objective (21) is to minimize the cost deviation between the recovery plan and the original plan. (Note that the objective is in fact the minimization of the costs of the recovery plan, because the costs of the original plan are constant.) Constraint (29) and (30) ensure vehicles leave from the central depot at their optimal starting times, where Kstak is the actual starting time of vehicle k from the depot and BSTk is its optimal starting time determined in Section 4.2; constraint (31) defines the range of some variables; constraint (32) ensures that virtual customers (Pseudo-depots) that are transformed from in-transit vehicles (Wang et al., 2009a; Li et al., 2009b) should be the first customers in the recovery plan, where VC is the set of virtual customers; other constraints own no difference in essence from the constraints in original VRPTW model except the representation of some parameters.

5. Nested partitions method for the recovery model We apply Nested Partitions Method (NPM) to solve the ´ lafsson (2000), is a proposed model. NPM, proposed by Shi and O novel global optimization heuristic algorithm which includes four operators: Partitioning, Sampling, Calculating the Promising Index and Backtracking. It was proved that the algorithm can converge to the global optimal solution with high probability. At present, NPM has been used in fields such as Traveling Salesman Problem (TSP), supply chain management, product design, machines sche´ lafsson, 2008). duling and so on (Shi et al., 2001; Shi and O Given a finite solution space and a performance function, NPM first determines a region (i.e., a subset of the finite solution space) as the most promising region; then this most promising region is partitioned into N subregions and the entire remaining region in the solution space is recognized as one more subregion. In each iteration, the N þ1 subregions are sampled randomly and their performances are calculated according to the performance function. The subregion with the best performance is selected as the new most promising region in the next iteration. If the current best solution is found in one of the N subregions, the subregion becomes the next most promising region; if the remaining region is found to be best, the algorithm will backtrack and a larger ´ lafsson, region becomes the next most promising region (Shi and O 2000). New iterations continue until the best solution is found. The designed NPM for the recovery model is as follows.

515

5.1. Partitioning Given a set of customers N ¼{1, 2, y, n} in the original routing and a set of new-adding customers Nnew ¼{1, 2, y, n_new}, N0 ¼{1, 2, y, n, n þ1,n þ2,y, n þn_new} represents the set of customers in the recovery routing. We apply customer-based coding scheme: {C_N, V_N, S_N} which means customer C_N is served by vehicle V_N and S_N determines the order of customers who are served by vehicle V_N. For example, scheme {3, 2, 1} represents customer 3 is served by vehicle 2 and customer 3 is the first customer of vehicle 2. The value ranges of above three variables are 0oC_Nr nþn_new, 0oV_NrK and 0oS_NrQ/min(di) where K is the number of vehicles in the depot, Q represents the maximum capacity of each vehicle and min(di) represents the minimum demand amount among all the customers, so the solution space is discrete and finite. Since the solution space is finite, the partitioning can be continued until all the subregions are singletons that cannot ´ lafsson, 2000). be partitioned further (Shi and O The Generic Partitioning method is used to partition the ´ lafsson, 2000). Given L¼[Q/min(di)] þ1 solution space (Shi and O represents the upper bound of S_N, we partition the solution space as follows: we firstly traverse the ranges of C_N (i.e. [1, nþ n_new]) in the coding scheme; then the solution space is divided by the values of V_N (i.e. [1, K]) until all the values are traversed; then each of the partitioned subregions is divided by the values of S_N (i.e. [1, L]) until all its values are traversed; lastly all the subregions are singletons. Fig. 9 illustrates the partitioning approach. Note that the values of ns in Fig. 9 are randomly chosen in their ranges when calculating the performance indexes and not all the values in the final subregions are feasible solutions.

Fig. 9. Partitioning the solution space.

Fig. 10. Sampling in the partitioned subregions.

516

X. Wang et al. / Int. J. Production Economics 140 (2012) 508–520

5.2. Sampling Assume that the Generic Partitioning is used and the current most promising region is in mth layer as Fig. 10 shows, this most promising region is partitioned into n subregions and the entire remaining region in the solution space is recognized as one more subregion. The n þ1 subregions are sampled and their performances are calculated according to the performance function which will be given in Section 5.3. If one of the subregions is found the best, the subregion will be the next most promising region. A proper sampling strategy of NPM will facilitate the searching for the best solution. The designed NPM for the combinational disruption recovery model integrates three rescue strategies: adding vehicles policy, neighboring rescue policy (Wang et al., 2009a; Li et al., 2009b) and starting later policy. (1) Adding vehicles policy. The strategy means that some new vehicles which have no delivery tasks according to the original plan are added to meet requests of new-adding customers. (2) Neighboring rescue policy. The strategy uses in-transit vehicles which adjoin newadding customers to deal with the disruptions. (3) Starting later policy. If vehicles do not leave from the depot until their optimal starting times, there may be some vehicles still staying at the depot when disruption events occur. Starting later policy is to make use of the vehicles which start later to meet requests of new-adding customers. According to the above three rescue strategies, the sampling method in the current most promising region is as follows. When disruption events occur, if there are vehicles waiting for their optimal starting times in the depot, the one with better promising index between starting later policy and neighboring rescue policy is chosen; if there are not vehicles waiting for their optimal starting times in the depot, the neighboring rescue policy is chosen. If both starting later policy and neighboring rescue policy cannot meet the changed requests, the adding vehicles policy is used. Finally, samples in the current most promising region are chosen. The samples in the remaining region are chosen by their performances in the last iteration. 5.3. Calculating the Promising Index The performance function of NPM should be determined to guide the searching of the algorithm. After samples are chosen, the possible recovery routing can be determined, so the service time deviation f1, the driving paths deviation f2 and the delivery costs deviation f3 from the original routing can be calculated. We take the inverse value of total deviation degree as the performance function:

t¼

1 f ðf 1 ,f 2 ,f 3 Þ

ð33Þ

where f ðf 1 ,f 2 ,f 3 Þ ¼ x1 f 1 þ x2 f 2 þ x3 f 3 , x1 þ x2 þ x3 ¼ 1. The smaller the service time deviation, the driving paths deviation and the delivery costs deviation of solutions are, the bigger their promising indexes are. In the current iteration, if the best value of the performance function is attained in some subregion of the current most promising region, the subregion will be taken as the next most promising region.

Fig. 11. Two ways of backtracking. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 4 Original data. Customers

X

Y

di

stai

endi

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

50 19 33 35 53 70 27 10 56 16 68 41 83 25 73 70

50 0 3 21 19 94 44 69 4 81 76 10 43 91 29 18

0 1.0 1.8 1.1 0.6 1.9 1.4 1.2 0.2 1.7 0.8 0.9 0.8 1.9 1.6 0.9

0 74 58 15 96 47 85 21 9 37 21 74 58 15 56 87

þN 144 128 85 166 117 155 91 79 107 121 174 158 125 156 187

region is one less than the depth of the current most promising region. The other condition is that the current best solution does not meet the constraints of the problem, i.e. an infeasible solution. Two ways of backtracking are applied: (1) to the parent layer of the current most promising region; (2) to the parent layer of the current best solution, as Fig. 11 shows. The red cross in Fig. 11 marks the backtracking point; the red branch represents the best solution in the current search space; the blue dotted line represents backtracking to the parent layer of the current most promising region; the pink dotted line represents backtracking to the parent layer of the current best solution. Under the worst conditions, the computational complexity is Ob, where O is the partitioning coefficient of the mth layer and b is ´ lafsson, 2000). Decreasing the the partitioning depth (Shi and O computational complexity and narrowing the solution space are two ways to increase the efficiency of NPM. We apply the three rescue strategies to narrow the solution space, which may reduce the elapsed time.

6. An example and computational experiments 6.1. Original VRPTW

5.4. Backtracking There are two main conditions under which the backtracking is needed. One is that better solutions than the current best solution are found in the remaining region. If the better index corresponds to the remaining region, the algorithm will backtrack to the parent region of the current most promising region. The depth of the parent

The original case of VRPTW studied in the paper is from Wang et al. (2009a): one depot owns 8 vehicles with the maximum capacity 6.5; the distance between two nodes can be calculated according to their coordinates; the average speed of each vehicle is 1; the fixed costs of dispatching a vehicle is 200; detailed original data can be seen in Table 4.

X. Wang et al. / Int. J. Production Economics 140 (2012) 508–520

By using improved Genetic Algorithm, Wang et al. (2009a) attained the optimal initial routing: 0-8-2-11-1-4-0; 0-10-5-13-0; 0-9-7-6-0; 0-3-14-12-15-0, as Fig. 12 shows. The total driving distance is 585.186.

6.2. Combinational disruption data After the initial scheduling, the dispatched vehicles left from the depot at time 0, and there are still 4 spare vehicles in the depot. At time 32.65, change requests are received from customer 4, 8, 11, 14 and a new customer 16 occurs. The detailed change data are shown in Table 5. When the above combinational disruptions happen, customers 3 and 10 have been visited by two of the dispatched vehicles and other two vehicles are in their ways to customers 8 and 9 respectively. Wang et al. (2009a), by applying the Rescheduling method whose objective was the minimum cost, produced the recovery routing: 0- virtual customer 1-8-6-0; 0- virtual customer 2-16-0; 0- virtual customer 3-13-9-7-0; 0- virtual customer 4-4-12-15-0; 0-5-0; 0-1-2-11-0; 0-14-0 and used Disruption Management by Genetic Algorithm (GA) to obtain the recovery routing: 0-virtual customer 1-8-2-0; 0- virtual customer 2-5-13-0; 0- virtual customer 3-9-7-6-0; 0- virtual customer 4-4-12-15-0; 0-16-1-0; 0-14-11-0. Both the above recovery routings produced by Wang et al. (2009a) are shown as Fig. 13. (In Figs. 13 and 14, the blue full lines are original routings and the red dotted lines are recovery routings; the blue circles and the red circles represent original customers and new-adding customers respectively; the green squares represent virtual customers transformed from in-transit vehicles.)

517

6.3. Disruption events transformation According to Section 4.1, the above disrupted nodes can be transformed into new-adding customer nodes: from 4, 14, 8, 11 to 16, 17, 18, 19 respectively, and the added node 16 is accordingly transformed into node 20, as Table 5 and Table 6 show. (Note that: original customers 4 and 14 will not be considered in the recovery routing after being transformed into new-adding customers, but customers 8 and 11 who only increase their demand should not be ignored.) If the dispatched vehicles depart from the depot at their optimal starting times: 22.84, 30.17, 31.57 and 52.35 which can be calculated according to Section 4.2, there will

Fig. 13. The recovery plans produced by the Rescheduling method (a) and Disruption Management based on GA (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 12. The original plan.

Table 5 Data of changes. Customers

Original coordinates

Original time windows

Original demand amount

New coordinates

New time windows

New demand amount

4 8 11 14 16

(53,19) (56,4) (41,10) (73,29) –

[96,166] [9,79] [74,174] [56,156] –

0.6 0.2 0.9 1.6 –

(53,29) Unchanged Unchanged Unchanged (55,60)

[10,54] Unchanged Unchanged [20,70] [30,75]

1.6 1.4 1.9 Unchanged 2.0

518

X. Wang et al. / Int. J. Production Economics 140 (2012) 508–520

be three in-transit vehicles traveling to their planed first customers and a dispatched vehicle has not left when the combinational disruptions occur at time 32.65. The three in-transit vehicles are transformed into virtual customers vc1, vc2, vc3. Data after the transformation are shown as Table 6. 6.4. Results and findings

Fig. 14. The recovery plans produced by NPM algorithm with neighboring rescue policy (a) and with starting later policy (b).

Table 6 Data after transformation. Customers

X

Y

di

stai

endi

Original customers

0 1 2 3 5 6 7 8 9 10 11 12 13 15 16 17 18 19 20

50 19 33 35 70 27 10 56 16 68 41 83 25 70 53 73 56 41 55

50 0 3 21 94 44 69 4 81 76 10 43 91 18 29 29 4 10 60

0 1.0 1.8 1.1 1.9 1.4 1.2 0.2 1.7 0.8 0.9 0.8 1.9 0.9 1.6 1.6 1.2 1.0 2.0

0 74 58 15 47 85 21 9 37 21 74 58 15 87 10 20 9 74 30

þN 144 128 85 117 155 91 79 107 121 174 158 125 187 54 70 79 174 75

Unchanged Unchanged Unchanged Unchanged Unchanged Unchanged Unchanged Unchanged Unchanged Unchanged Unchanged Unchanged Unchanged Unchanged 4 14 8 11 The new customer 16

Note that: The italicized data are the changed data; original customers 4 and 14 are not considered in the recovery routing after being transformed into newadding customers, but customers 8 and 11 who only increase their demand should not be ignored.

We programmed the designed Nested Partitions Method for the proposed recovery model in Cþþ language with Matlab software. According to the actual situation, coefficients y, s and m are assigned 1, 1 and 10 respectively. And goal weights x1 , x2 and x3 are 0.4, 0.3 and 0.3 respectively. NPM algorithm with neighboring rescue policy produced the new routing: 0- virtual customer 1-17-18-8-2-0; 0- virtual customer 2-10-5-13-0; 0- virtual customer 3-9-7-6-0; 0-3-1-19-11-16-12-0; 0-20-15-0, as Fig. 14(a) shows. NPM algorithm with starting later policy produced the new routing: 0- virtual customer 1-8-2-11-1-0; 0- virtual customer 2-10-5-13-0; 0- virtual customer 3-9-7-6-0; 0-16-3-18-12-15-0; 0-20-17-19-0, as Fig. 14(b) shows. The results compared with Wang et al. (2009a) are shown as in Table 7. From the comparison, we find: (1) Disruption Management by GA is superior to the Rescheduling method in Paths deviation, Service time deviation and Number of vehicles; Disruption Management by NPM with neighboring rescue policy produces better results in Total distance, Paths deviation, Service time deviation and Number of vehicles than the Rescheduling; Disruption Management by NPM with starting later policy also outdoes the Rescheduling method in all aspects. These verify the advantage of Disruption Management in dealing with disruption events for VRPTW. (2) In Total distance and Number of vehicles, Disruption Management by NPM with neighboring rescue policy is superior to the Rescheduling and Disruption Management by GA; Disruption Management by NPM with starting later policy produces better results than the Rescheduling and Disruption Management by GA in all aspects. These give some evidences on that the proposed transformation methods of disruption events and the designed NPM algorithm in this study are effective to solve the combinational disruption recovery model. (3) Disruption Management by NPM with starting later policy is better than Disruption Management by NPM with neighboring rescue policy in Paths deviation and Service time deviation, which hints that considering vehicles’ optimal starting times from the depot may provide a new rescue strategy for disrupted VRP. We change the values of goal weights x1 , x2 and x3 when calculating the promising index in NPM algorithm to analyze their impacts on the recovery plans, as Table 8 shows. As we can see, the goal weights take a crucial part on the results: (1) Service time deviation decreases as its weight x1 increases, which results in the increase of Total distance, and Paths deviation has the same trend; (2) the increase of delivery costs deviation weight x3 produces shorter Total distance but brings about bigger Paths deviation and Service time deviation. Thus, it seems impossible to reduce Service time deviation, Paths deviation and Total distance simultaneously when producing recovery plans for disrupted VRP. How on earth decision-makers will do may depend on their concerns among customers’ satisfaction, drivers’ feeling and delivery costs.

7. Conclusions There are various unexpected disruption events encountered in the delivery process. These disruptions often make actual delivery operations deviate from intended plans, which may bring different disturbances on the real-world participators (customers, drivers and providers) in logistics delivery. Although the rescheduling

X. Wang et al. / Int. J. Production Economics 140 (2012) 508–520

519

Table 7 Comparison of results. Methods

The routings

Total distance

Paths deviation

Service time deviation

Number of vehicles

The original routing

0-8-2-11-1-4-0 0-10-5-13-0 0-9-7-6-0 0-3-14-12-15-0

585.186

–

–

4

Rescheduling

0-vc1-8-6-0 0-vc2-16-0 0-vc3-13-9-7-0 0-vc4-4-12-15-0 0-5-0 0-1-2-11-0 0-14-0

840.76

29

210.75

7

Disruption Management by GA

0-vc1-8-2-0 0-vc2-5-13-0 0-vc3-9-7-6-0 0-vc4-4-12-15-0 0-16-1-0 0-14-11-0

841.69

19

37.03

6

Disruption Management by NPM with neighboring rescue policy

0-vc1-17-18-8-2-0 0-vc2-10-5-13-0 0-vc3-9-7-6-0 0-3-1-19-11-1612-0 0-20-15-0

679.79

20

66.76

5

Disruption Management by NPM with starting later policy

0-vc1-8-2-11-1-0 0-vc2-10-5-13-0 0-vc3-9-7-6-0 0-16-3-18-12-15-0 0-20-17-19-0

737.11

14

33.3

5

Note that: vci that is short for virtual customer i in the table is ignored when calculating the paths deviation.

Table 8 The impacts of goal weights on results of Disruption Management by NPM with starting later policy.

x1

x2

x3

Total distance

Paths deviation

Service time deviation

Number of vehicles

0.2 0.4 0.6 0.8 0.5 0.4 0.3 0.2 0.5 0.4 0.3 0.2

0.4 0.3 0.2 0.1 0.1 0.3 0.5 0.7 0.4 0.3 0.2 0.1

0.4 0.3 0.2 0.1 0.4 0.3 0.2 0.1 0.1 0.3 0.5 0.7

706.32 737.11 756.38 823.29 711.16 737.11 748.70 811.69 826.30 737.11 718.12 701.35

11 14 16 21 18 14 12 10 13 14 18 24

48.22 33.30 23.76 18.27 28.76 33.30 43.90 46.78 31.25 33.30 45.74 48.50

5 5 5 6 5 5 5 6 6 5 5 5

method can produce relatively economic rescue plans for disruption events of vehicle routing problems, it may bring great disturbance to the whole delivery system. Disruption Management provides a good idea to minimize the negative effects on the participators in logistics delivery. Most existing researches for disrupted VRP mainly focus on producing new routings with the minimum costs, ignoring the real-world participators. For delivery disruption events, there may be several recovery alternatives, but different participators may prefer different alternatives. It is important to consider the satisfaction of the participators which has important effects on the long-term development of delivery firms when working out recovery plans for disrupted VRP. We measure the new-adding

customer disruption from aspects of customers, drivers and logistics providers, and applies the thought of Disruption Management to produce solutions which are satisfactory to the three participators. For the reality that a variety of delivery disruptions often occur successively or simultaneously, we propose methods of transforming various disruption events into new-adding customer disruption, which facilitates to develop a common and combinational VRPTW disruption recovery model. Considering vehicles’ optimal starting times from the central depot can not only reduce the waiting costs of in-transit vehicles but also provide a new rescue strategy for the disrupted VRP. Nested Partitions Method (NPM) is used to solve the proposed recovery model. We focus on various customer disruption events in computational experiments, giving no consideration to vehicle disruption events and cargo disruption events which need further efforts. One of next works is to develop more effective and efficient multiobjective optimization algorithms for the proposed combinational disruption recovery model, which will be helpful to developing a decision support system for various disruption events of delivery practices.

Acknowledgment We gratefully acknowledge the anonymous referees for their constructive comments on the manuscript. We are so thankful to Professor Edwin Cheng for his valuable suggestions and his assistant Julia Yam for her helpful work. This work is supported by the National Natural Science Foundation of China (nos. 90924006, 71171029, 70890080, and 70890083) and the National Natural Science Funds for Distinguished Young Scholar (no. 70725004).

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