Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 2096416 |

Li Jiang, Changyong Liang, Junfeng Dong, Wenxing Lu, Marko Mladenovic, "A Disruption Recovery Problem with Time Windows Change in the Last Mile Delivery of Online Shopping", Mathematical Problems in Engineering, vol. 2018, Article ID 2096416, 10 pages, 2018.

A Disruption Recovery Problem with Time Windows Change in the Last Mile Delivery of Online Shopping

Academic Editor: Adrian Chmielewski
Received29 Aug 2018
Revised15 Nov 2018
Accepted02 Dec 2018
Published10 Dec 2018


Frequent time window changing disruptions result in high secondary delivery rates in the last mile delivery. With the rapid growth of parcel volumes in online shopping, the time window changing disruptions could translate to substantial delivery cost-wastes. In recent years, customer pickup (CP), a new delivery mode that allows customers to pick up their parcels from shared delivery facilities, has provided a new way to deal with such disruptions. This study proposed a disruption recovery problem with time windows change in the last mile delivery in which customers can be served through home delivery (HD) or CP. A variant variable neighborhood descent (VVND) algorithm was presented to solve the problem. Computational experiments based on a set of instances were tested, and results were compared with other heuristics in the literature, which have affirmed the competitiveness of the model and algorithm.

1. Introduction

The rapid e-commerce growth results in a fast increase of parcel delivery. According to the State Post Bureau of China, the parcel volume of online shopping of 3.67 billion in 2011 increased to 40.06 billion in 2017, with an average growth rate of more than 48% within six years. This increase of parcel volume also caused the increase of the one-time delivery failure rate. Many parcels failed to be delivered at the first attempt [1], and these parcels had to be delivered the second time or even the third time. The main reason for this high delivery failure rate was the “not-at-home” problem (customers are going out); thus, the agreed delivery time between a courier and a customer commonly changes. Reducing the high delivery failure rate to deal with the frequent time window changing disruptions had become a key issue in the last mile delivery of online shopping.

Previously, in terms of the last mile delivery, home delivery (HD) (that is, delivering parcels directly to customers’ homes or workplaces) was common. Recently, customer pickup (CP) has become widely popular, because it allows customers to pick up their parcels from shared delivery facilities (SDFs) near their homes or workplaces at their convenience. The SDFs widely used in more than 80 cities in China include the “CaiNiao” station established by Alibaba Company, “FengChao” established by SF-Express, and “Sposter” established by the China Post Group.

The rescheduling of delivery routes [2, 3] plays a good role in solving the disruption recovery problem (DRP) in the HD mode, which is an NP-hard problem [3]. Meanwhile, for the DPR in the CP mode, deciding on the allocation of SDFs while rescheduling the delivery routes is important. The DRP in the CP mode is more complicated than that in the HP mode. Although many articles were published on the DRP in the HD mode [24], only a few studies explored the DRP in the CP mode. This study proposed a DRP with time windows change in the last mile delivery to solve the practical problems and bridge the gap in the literature. The customers could be served either by couriers (HD) or by SDFs (CP). The purpose was to decide on the allocation of SDFs and rescheduling of the delivery route to minimize the total delivery cost when a disruption (time windows changed) would occur. A variant variable neighborhood descent (VVND) algorithm was proposed to solve this problem.

The rest of this paper is structured as follows. Section 2 provides a comprehensive review of the related studies. Section 3 details the description and formulation of the proposed DRP. Section 4 discusses the proposed VVND algorithm. Section 5 displays the results of the computational experiments and the discussion. Section 6 concludes this paper.

2. Literature Review

One key issue related to our research was the shared delivery facilities (parcel lockers or shared reception boxes) in the last mile delivery. The SDF studies were categorized into two streams: case study and routing optimization.

The case studies analyzed the benefits of the SDFs in the last mile delivery by using survey data. Punakivi et al. [5] emphasized that approximately 55%-66% of cost could be reduced from home delivery by allowing customers to pick up parcels from shared reception boxes. A similar study by Kämäräinen [6] contended that the cost reduction was 42%. Edwards et al. [7] confirmed that collecting parcels from parcel lockers could reduce carbon emission by 83%. Morganti et al. [8] focused on the self-collection networks in Europe and affirmed that the parcel lockers could reduce the operational cost of last mile logistics delivery. Meanwhile, Lemke et al. [9] analyzed the usability of the parcel lockers from customers’ perspectives by using data from Polish cities, and the results validated that most consumers prefer using lockers near their homes. Yuen et al. [10] explored customers' intention to use self-collection in the last mile delivery by using data from 164 consumers located in Singapore.

The routing optimization focused on solving the SDF location and the delivery routing problem. Mainly, Deutsch et al. [11] studied the location allocation problem of parcel lockers in the last mile delivery and aimed to maximize the profit of a consignment company by deciding on the location of parcel lockers. Zhou et al. [12] proposed a location-routing problem with home delivery and customer pickup for the city distribution of online shopping to minimize the vehicle, routing, opening of SDF, and second delivery costs in the last mile delivery. Zhou et al. [13] also introduced a two-echelon vehicle routing problem with delivery options in the last mile distribution, wherein customers could be served by home delivery or customer pickup to minimize the total distribution cost.

Another key issue related to our research is the disruption recovery problem (DRP), which is involved in multiple research areas, such as the passenger transport disruption recovery problem [1416], vessel schedule recovery problem [17, 18], and logistics distribution recovery problem. In logistics, Zeimpekis et al. [19] introduced a management system framework for dealing with disruption accidents in urban logistics distribution. Protvin et al. [20] studied the disruption accidents of new customers and uncertain travel time that were frequently encountered during the collection of tasks. Mu et al. [4] proposed a disrupted vehicle routing problem (VRP) to deal with the disruptions that occur at the execution stage of a VRP plan, while Zhang el al. [21] analyzed the effect of disruptions on road networks and the recovery process by using domain wall theory.

Existing literature on the DRP had produced effective solutions for various disruptions, especially the disruption that occurred in logistics. However, such literature mainly focused on the DRP in the HD mode, and no study on the disruption problem in the CP mode was available. This study explored the disruption recovery problem (DRP) with time windows change in the last mile delivery in the CP mode to solve the practical problem and bridge the gap in the literature.

3. Problem Formulations

3.1. Problem Description

A network without SDFs in the last mile delivery is described as follows.  , , stands for depot, and stands for customer set. , is a set of edges connecting each pair of nodes in . The travel cost and travel time of the edge in from to are and , respectively. The time window of is expressed as . Figure 1 exhibits the network and initial route.

A network with SDFs in the last mile delivery is described as follows., , is the depot set; is the customer set, and is the SDF set. stands for the changed time window of customer . , is a set of directed edge connecting each pair of nodes in ; the delivery distance and time of the directed edge in are and , respectively. contains the edges between SDF and customer , and the pickup distance and the time of the edge in are and , respectively. The unit opening cost of SDF is . Figure 2 presents the network with SDFs.

Couriers depart from depots and deliver parcels according to the initial optimal solution. During delivery, couriers must adjust the delivery route to obtain a good solution when a disruption occurs (customers’ time windows changed). In this paper, the objective function is to minimize the cost related to customer satisfaction, route cost, and SDF opening cost.

Without losing generality, the following assumptions are made.

The capacity limit of SDFs is not considered in this paper.

Customer satisfaction is only inversely proportional to the delivery delay time and customer pickup distance.

Couriers must wait until to start the service of customer ; they can deliver parcels to SDFs at any time.

Each customer can be served by either couriers or SDFs.

3.2. Notions and the Proposed Model

Table 1 summarizes the notions used in the formulation.


VNode set,
Depot set,
Customer set,
SDF set,
EArc set,
Delivery arc set,
Pickup arc set,


MNumber of SDFs
NNumber of customers
Distance of arc
Travel time of arc
Pickup cost of arc
Pickup time of arc
Time window of customer
Unit opening cost of SDF p
LLarge positive number
Sub-weight of delivery delay time
Sub-weight of pickup distance
Weight of pickup cost
Weight of routing cost
Weight of SDFs opening cost
Server radius of the SDFs

Decision variables

Binary variable, , if arc(i,j) is used by courier; otherwise,
Binary variable, , if customer is assigned to SDF ; otherwise,
Integer variable, arrival time to node

When a disruption occurs, the influence of customers and couriers is different.

The influence of disruption on customer is service satisfaction. According to assumption (2), service satisfaction is related to delivery delay time and customer pickup distance. Service satisfaction decreases with the increase of delivery delay time and customer pickup distance, which are shown in Figures 3 and 4, respectively.

The cost related to service satisfaction can be formulated as Formula (1).

The influence of disruption on couriers is delivery route and SDF opening costs, the function related to delivery distance is shown as Formula (2), and the function related to SDF opening cost is shown as Formula (3).

On the basis of the above statements, the model for the proposed problem can be stated as follows:

which is subject to

Objective function (4) minimizes the sum of the cost related to service satisfaction, route cost, and SDF opening cost. Constraints (5) and (6) ensure that each customer is served exactly once by either SDF or a courier, and once a courier enters a node, it must also leave for it. Constraint (7) ensures that the SDF that is allocated for customers must be allocated to the route. Constraints (8) and (9) guarantee that the working time of customer should be satisfied. Constraint (10) makes sure that the courier must be served only after the time window has started. Constraint (11) makes sure that the service radius of the SDF is within the allowable distance. Constraints (12) to (13) define all variables.

4. Hybrid Approach

For the disruption recovery problem, the algorithm should focus not only on solution accuracy, but also on solution speed. If the solution time is too long, then the solution may have been infeasible. The variable neighborhood descent (VND) has been successfully applied for solving hard combinatorial optimization problems, and it particularly performs well for solving LRPs [22, 23]. This study proposes a variant variable neighborhood descent (VVND) algorithm for solving the disruption recovery problem. The innovation of the VVND includes two aspects. Two new operators (addBox() and dropBox()) are added in the proposed algorithm. The 2-opt operator is improved by using the location range (LR) to speed up convergence. Algorithm 1 presents an overview of the VVND.

4rd = rand();
5if rd < 0.5
7 S’ = addBox(S, openNum);
9 closeNum = rand()%S.openBoxNum;
10 S’ = dropBox(S, closeNum);
12S’ = FindBestNeighbor(S’);
13if f(S’) < f(S)
14S = S’;
15k = 1;
17if iter < iterMax
18 iter = 0;
19 k = k + 1;
21 end
22iter = iter + 1;
23Until k > kMax
4.1. Solution Representation

A mixed encoding scheme is designed to represent a solution. In the encoding scheme, two sections are shown in Figure 5.

First section is an assignment used to label each customer's service options. If the value in the assignment belongs to , then the customer picks up the parcel from the SDF. If the value is 0, then the customer receives the parcel at home or workplace.

Second section is a route consisting of customers and SDFs. In the routing, courier starts and ends at the depot and delivers parcels to customers or SDFs one by one.

4.2. addBox() and dropBox() Procedures

The steps of the addBox() are as follows:(1)Randomly select openNum closed SDFs (if there are any) and open them.(2)Reallocate the customers within the service radius of the SDFs.(3)Randomly insert the SDFs into the route.

The steps of dropBox() are as follows:(1)Randomly select closeNum opened SDFs (if there are any).(2)Delete the SDFs from the delivery route.(3)Randomly insert the customers of the SDFs into the delivery route.

Figure 6 illustrates the addBox() and dropBox().

4.3. Neighborhood Structures

Insertion move (N1). The insertion move is carried out by selecting everyach two nodes, that is, node and nodes of the solution, and reinserting the node into the position immediately before (or in) the node of the solution. The insertion move has two effects as follows.

Case (a): If the selected node is a customer (in the route or SDF), then it can be reinserted to the route or the SDF.

Case (b): If the selected node is a SDF, then it can be only reinserted to route.

Figure 7 illustrates the insertion move.

Improved 2-Opt move (N2). A subsequence is obtained by selecting each two nodes, that is, the ith and jth nodes of the solution. Reversing the order of the subsequence can obtain a new solution. Note that the selected node ith or jth cannot be a customer in SDF. Figure 8 illustrates the 2-opt move.

In the network with time windows, partial arcs are disconnected due to the time window constraints, which can be used to calculate each node location range (LR) in the route. By using LR, partial useless exchange can be reduced.

Figure 9 exhibits that the location range of customer is , the useless exchange in can be deleted, and the can be calculated by Formula (14).

5. Computational Experiments

The proposed algorithm is compiled with C++ and runs on PC with an Intel i5-7500 3.40 GHz CPU, 8.00GB RAM.

5.1. Test with Time Windows Change

The test dataset consists of 100 customers (an instance n100w60.004 proposed by Dumas et al. [24]) and nine SDFs. The coordinates of the SDFs are evenly distributed in the area of ,: 102(10,10), 103(10,30), 104(10,45), 105(25,10), 106(25,30), 107(25,45), 108(40,10), 109(40,30), 110(40,45). The known optimal solution is , 94, 58, 42, 98, 82, 9, 59, 39, 75, 53, 35, 13, 84, 41, 5, 50, 70, 80, 25, 47, 55, 66, 95, 11, 23, 2, 100, 30, 67, 31, 26, 96, 21, 81, 79, 27, 38, 17, 61, 92, 86, 57, 15, 36, 19, 85, 77, 60, 20, 69, 44, 48, 64, 40, 52, 72, 73, 7, 32, 87, 29, 90, 22, 51, 65, 56, 49, 46, 91, 54, 71, 74, 78, 43, 68, 12, 18, 33, 14, 37, 88, 8, 10, 93, 16, 99, 3, 83, 97, 62, 89, 28, 45, 76, 24, 6, 4, 101, 34, , and the route cost is 764. When the courier delivers to the customer 98#, the courier receives the time window change information (Table 2).

Customer no.Status Time windowCustomer no.Status Time window

6Old TW62766443Old TW382413
New TW50100New TW3080
10Old TW46049947Old TW34122
New TW100150New TW500540
13Old TW216753Old TW068
New TW400440New TW600668
27Old TW14523362Old TW510574
New TW350400New TW110170
34Old TW72579570Old TW37128
New TW200260New TW700758

The parameters are the same as the above tests: , , , , . The optimal solution obtained by the proposed algorithm is shown in Tables 3 and 4. Table 3 presents the customer assignments of the SDFs, and Table 4 presents the delivery route.

SDF no.Customer no.SDF no.Customer no.



This study reports the comparisons of the reschedule route in the CP mode (RRCP) with the original delivery route (ODR) and the reschedule route in the HD mode (RRHD), where Difference = RRCP – RRHD and GAP = (RRHD - RRCP) / RRHD 100 (see Table 5).


Delay time5300052117−50496.74%
Pickup cost00169+169
Opening cost00105+105
Route cost764703442−26137.13%
Total cost537641224733−49140.08%
CPU Time (S)02122

Compared with the RRHD, the RRCP increases the pickup cost and SDF opening cost by providing customer pickup services, but the total cost is reduced by 40.08%. The delay time is reduced from 521 to 17 or 96.74%; route cost is reduced from 703 to 442 or 37.13%. As carbon emission is closely related to delivery route distance, the proposed model can help effectively reduce carbon emission.

5.2. Sensitive Analyses

To gain additional insights, a sensitive analysis is conducted to show how the acceptable distance influences cost. When ranges from 0 to 50, the results are reported in Table 6 and Figure 10.


Delay time521197971717171717
Pickup cost04197169169169169169
Opening cost06080105105105105105
Route cost703647552442442442442442
Total cost1224944826733733733733733

From the sensitive analysis, we can see that when , the proposed method is optimal without SDF. When , the delay time and route cost are first to fall continuously and then keep invariability. The pickup cost is increased with the increase of . From a cost optimization perspective, is the best SDF service radius in this instance.

5.3. Computational Results Comparison

For the evaluation of the performance of the proposed algorithm, tests based on the benchmark instance proposed by Dumas et al. [24] are used. The networks consist of one depot and 20, 40, 60, 100, and 150 customers. The width of time window is 60. The locations of SDFs are , (14,28), (48,49), (8,49), (48,25), (41,8), (22,46), (40,48), (33,2), (43,47), (34,38), (38,20), (33,9), (36,2), (14,3), (5,42), (35,16), (48,2), (22,20), (39,40), (10,25), (23,33), (36,38), and , which are generated randomly in an area of ,×,. The parameters are listed as follows: , ,,, . The parameters of VVND are kMax = M and iterMax = M/2.

5.3.1. Comparison with Other Algorithms

Table 7 presents comparisons of the proposed VVND and Simulated Annealing (SA). N represents the number of customer nodes; M represents the number of SDFs. The values under the “GAP” columns are calculated by the formula []/100. The solutions obtained by SA [25] are shown in Columns 4-5. Columns 6-7 show the solutions obtained from the proposed VVND. Time is the CPU time in seconds on a PC (i5-7500@3.40 GHz, 8.00GB RAM). The parameters of SA are set as follows: T=N, alpha=0.98, and =M. All the best solutions of SA and VVND are obtained with 5 runs of the algorithms.



Table 7 indicates that the proposed algorithm outperforms SA in terms of solution quality. The VVND produces the best solutions to all 25 instances, while SA produces only 8 best solutions. The average gap between the two algorithms is 1.87%.

5.3.2. Comparison with HD

Table 8 exhibits the results. N represents the number of customers; M represents the number of SDFs; GAP = (Total cost1-Total cost2) / Total cost1100. The values of column “Total cost1” are the best known solution proposed by Dumas et al.[24]. The values of column “Total cost2” are the best solution obtained by 5 runs.

No. N MHD Proposed methodGAP


Table 8 shows the following. Compared with HD, the proposed method has a significant improvement, and the total cost is reduced by 3.44% to 34.82% with the increased SDFs. When the service radius of the SDF is fixed, with the increased SDFs, pickup and the opening costs increase, but the total cost decreases.

6. Conclusions

This study proposed a disruption recovery problem with time window changes in the last mile delivery of online shopping. One characteristic of the problem is that the delivery route could be rescheduled by using SDFs when time window changing disruptions occurred during the delivery process. The VVND algorithm was presented, which included a 2-opt algorithm improved by LR to speed up its convergence, to solve this problem. The proposed method and algorithm were tested on a set of instances. The results corroborated that the proposed method could quickly recover the delivery and reduce the total delivery cost when time window changing disruptions occurred in the last mile delivery. Meanwhile, the proposed algorithm had an evident competitiveness compared with other heuristics, and the total delivery cost could be reduced by approximately 3% to 34% with the increasing number of SDFs compared with the solution of the HD mode.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


This work was supported by the National Science and Technology Support Program of China (Grant Nos. 71331002, 71502047, 71601061, and 71771077); the Ministry of Chinese Education, Humanities and Social Sciences project (Grant No. 17YJA630037); the National Key R&D Program of China (No. 2016YFC0803203); the Fundamental Research Funds for the Central Universities project (Grant No. JS2017HGXJ0044); the “Double-First Class” Construct Project (Grant No. 45000-411104/005); and the Natural Science Foundation of Anhui Province in China (Grant No. 1808085QG229).


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