Journal of Advanced Transportation
Volume 2021 (2021), Article ID 5182989, 22 pages
https://doi.org/10.1155/2021/5182989

Research Article

Multi-Depot Pickup and Delivery Problem with Resource Sharing

Yong Wang,¹ Lingyu Ran,¹ Xiangyang Guan,² and Yajie Zou³

¹School of Economics and Management, Chongqing Jiaotong University, Chongqing 400074, China
²Department of Civil and Environmental Engineering, University of Washington, Seattle, WA 98195, USA
³Key Laboratory of Road and Traffic Engineering of Ministry of Education, Tongji University, Shanghai 201804, China

Correspondence should be addressed to Yajie Zou; zouyajie@tongji.edu.cn

Received 10 April 2021; Revised 28 April 2021; Accepted 15 May 2021; Published 2 June 2021

Academic Editor: Xiaoyue Liu

Copyright © 2021 Yong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Resource sharing (RS) integrated into the optimization of multi-depot pickup and delivery problem (MDPDP) can greatly reduce the logistics operating cost and required transportation resources by reconfiguring the logistics network. This study formulates and solves an MDPDP with RS (MDPDPRS). First, a bi-objective mathematical programming model that minimizes the logistics cost and the number of vehicles is constructed, in which vehicles are allowed to be used multiple times by one or multiple logistics facilities. Second, a two-stage hybrid algorithm composed of a k-means clustering algorithm, a Clark-Wright (CW) algorithm, and a nondominated sorting genetic algorithm II (NSGA-II) is designed. The k-means algorithm is adopted in the first stage to reallocate customers to logistics facilities according to the Manhattan distance between them, by which the computational complexity of solving the MDPDPRS is reduced. In the second stage, CW and NSGA-II are adopted jointly to optimize the vehicle routes and find the Pareto optimal solutions. CW algorithm is used to select the initial solution, which can increase the speed of finding the optimal solution during NSGA-II. Fast nondominated sorting operator and elite strategy selection operator are utilized to maintain the diversity of solutions in NSGA-II. Third, benchmark tests are conducted to verify the performance and effectiveness of the proposed two-stage hybrid algorithm, and numerical results prove that the proposed methodology outperforms the standard NSGA-II and multi-objective particle swarm optimization algorithm. Finally, optimization results of a real-world logistics network from Chongqing confirm the applicability of the mathematical model and the designed solution algorithm. Solving the MDPDPRS provides a management tool for logistics enterprises to improve resource configuration and optimize logistics operation efficiency.

1. Introduction

With the advancement of information technology and Internet of Things, the logistics industry is playing an increasingly important role in the development of modern businesses [1, 2]. However, national and local governments worldwide are focusing on the environmental impacts of logistics and the efficient use of resources [3, 4]. In a logistics network, customers send out a series of requests for delivery and pickup services, and logistics service providers (LSPs) design service plans and arrange vehicles for these requests to deliver or pickup goods [5, 6]. Efficient logistics service plan can improve the operation efficiency of LSPs and resource utilization [7, 8]. Therefore, making an effective logistics service plan with resource sharing (RS) is essential, which not only helps to reduce the operating cost for logistics facilities but also promotes the development of green logistics and provides better logistics services for consumers [9, 10].

In this study, a multi-depot pickup and delivery problem with RS (MDPDPRS) combines components from three subproblems: multi-depot vehicle routing problem (MDVRP) with pickups and deliveries (MDVRPPD), MDVRP with pickups and deliveries and time windows (MDVRPPDTW), and RS [11–13]. However, one of the difficult challenges in solving this problem is how to handle pickup and delivery activities among multiple depots through RS [14, 15]. In the traditional MDVRPPD, each vehicle performs only one type of activity in the service route, which may be delivering or picking up goods [16–18]. In addition, the traditional MDVRPPDTW and single-depot vehicle routing problem with pickups and deliveries mostly only consider the optimization of logistics operational costs [19, 20]. Therefore, MDPDPRS focuses on how to support and achieve the efficient utilization of transportation resources with RS strategy, and optimizes the logistics network.

With regard to RS, it is often jointly adopted with collaboration or cooperation between LSPs to optimize the logistics networks with multiple depots [21, 22]. RS strategy supports the sharing of customer information and transport resources to improve the resource configuration among logistics facilities to optimize the logistics network [23, 24]. Here, the sharing of customer information is often enabled by customer clustering, whereas the sharing of transportation resources is related to the use of shared transportation equipment [25, 26].

As for MDVRPPD, it is a crucial logistics issue with extensive applications, especially in reverse logistics [27]. Three basic types of vehicle routing problems exist in reverse logistics [28]. The first type is the vehicle routing problem with mixed deliveries and pickups, which involves customers with delivery demands, pickup demands, and delivery and pickup demands [29]. Vehicle routing problem with simultaneous delivery and pickup is the second type, which requires all customers to have both delivery and pickup demands [30]. The third type is the common MDVRPPD, which includes delivery and pickup customers in the logistics network [19, 31–33]. In this study, the consideration of customer service time windows makes MDVRPPD realistic.

In this study, the MDPDPRS can be formulated into a bi-objective mathematical model to minimize the total logistics operating cost and number of vehicles [32, 34]. On the basis of the multi-depot and RS properties of MDPDPRS, a two-stage hybrid algorithm is proposed to find the Pareto optimal solution. In the first stage, a k-means algorithm is adopted to reconfigure resources through customer clustering; thus, the MDVRPPDTW is simplified for solving [35]. The second stage focuses on finding the Pareto optimal solution for the bi-objective optimization problem [23, 36]. The Clarke-Wright (CW) algorithm, which is good at constructing the initial solution of vehicle routes, and the nondominated sorting genetic algorithm (NSGA-II), which is known for its capability of finding the Pareto solution, are adopted to optimize the vehicle routing in the second stage [33, 37, 38].

The remainder of this study is arranged as follows. Section 2 reviews the relevant literature. Section 3 elaborates the specifics of the MDPDPRS. Section 4 explains the bi-objective mathematical model for the MDPDPRS. Section 5 presents the designed methodology for solving the MDPDPRS. Section 6 analyzes the performance and application of the proposed model formulation and solution algorithm in a real-world case study compatible to the MDPDPRS. Finally, Section 7 summarizes the conclusions and discusses potential future research.

2. Literature Review

MDPDPRS is mainly related to the MDVRPPD, MDVRPPDTW, and RS strategy [18, 36]. MDVRPPD and MDVRPPDTW are the extension problems of MDVRP and MDVRPTW with respect to the logistics service type of customer demands, responsively [39, 40]. In the widely studied MDVRP and MDVRPTW, the service types of customers are either deliveries or pickups [16, 33, 41]. However, in a real-world logistics network with multiple depots, customers with distribution and pickup demands often exist simultaneously and the service time windows are the additional characteristics of customers; this issue is abbreviated by scholars as MDVRPPD [11, 13]. Therefore, MDVRPPD and MDVRPPDTW have begun to attract the attention of scholars, and the difference between the two issues is mainly whether customers’ service time window feature is considered [42, 43]. In the MDVRPPDTW, the optimization of vehicle routes focuses on the confirmation of customers’ service time windows and the integration of vehicles’ delivery and pickup activities, which are suitable with the factors that LSPs must consider if optimizing vehicle routes [44, 45]. The adoption of RS into the optimization of logistics network is also a current trend [12]. Thus, the MDPDPRS in this study integrates vehicles’ distribution and pickup arrangement with the RS strategy to optimize the MDVRPPDW.

In contrast to the MDVRP, the MDVRPPD studies customers with delivery and pickup demands [35, 42]. The MDVRPPDTW is an extension of MDVRPPD, which considers the characteristics of customer service time windows [46, 47]. Many scholars have studied MDVRPPD and MDVRPPDTW considering diverse aspects and proposed different mathematical models and algorithms [30, 48]. In terms of models, many of the proposed mathematical models reflect the characteristics of their problem studied by different constraints, including capacity, time windows, and priority constraints [29, 49]. Ropke et al.[50] established a standard three-index model based on the characteristics of customer time window and designed an accurate algorithm to solve it. Gribkovskaia et al.[19] studied the vehicle routing problem with deliveries and pickups considering the number of times a customer has been visited and proposed a mixed integer linear programming model. Chen et al.[31] established a comprehensive mathematical model to minimize the transportation costs for unpaired vehicle routing problem with deliveries and pickups in a multi-factory production network. Conversely, hybrid heuristics algorithms (e.g., genetic algorithm and adaptive large neighborhood search algorithm) and exact algorithms based on column generation are commonly designed to solve MDVRPPD and MDVRPPDTW [50–52]. The proposed model and solution methodology in the above studies provide abundant reference for solving the basic MDVRPPD and MDVRPPDTW. However, few studies have optimized MDVRPPDTW with RS strategy [32, 34].

Customer clustering analysis is a research aspect that groups customers based on their characteristics (e.g., location and time window), and common clustering algorithms include k-means clustering, parallel clustering, and fuzzy-based customer clustering [53–56]. In comparison with other customer clustering algorithms, k-means clustering is widely adopted to solve vehicle routing problems [57, 58]. Xu et al.[59] proposed an enhanced ant colony algorithm based on k-means clustering to solve dynamic vehicle routing problems and achieved good optimization results. Hakim et al.[60] designed a cluster-based method to solve a vehicle routing problem with limited vehicle capacity, and their calculation results proved the effectiveness of that method. Mourelo Ferrandez et al.[61] reduced the calculation difficulty of truck-drone in tandem delivery network by k-means clustering algorithm. Wang et al.[35] proposed a hybrid heuristic algorithm based on three-dimensional k-means clustering and improved reference point NSGA-II to solve the multi-objective optimization model. Therefore, cluster analysis can simplify the difficulty of solving vehicle routing problem [62, 63].

At present, the construction of multi-objective optimization model and multi-objective optimization algorithm is the research hotspot of finding the Pareto solution of vehicle routing problems, and scholars have designed different algorithms [38, 64]. NSGA-II and multi-objective particle swarm optimization (MOPSO) algorithm are two common multi-objective algorithms [65, 66]. NSGA-II adopts a reference point strategy to maintain population diversity [67]. Srivastava et al. [68] proposed a NSGA-II to solve the multi-objective optimization model for MDVRPTW and verified that the method is superior to the latest method of that problem through a real-world case study. Maadanpour Safari et al.[69] adopted NSGA-II, multi-objective simulated annealing (MOSA), and MOPSO to optimize the proposed three-objective mathematical function and concluded that NSGA-II was superior to MOSA according to the results of their examples. Shafiei Nikabadi et al. [70] formed a multi-objective model for the route selection of freight fleet, optimized it with NSGA-II and MOPSO, and considered that MOPSO was superior to NSGA-II. Therefore, NSGA-II and MOPSO are two commonly typical multi-objective optimization algorithms that can find Pareto solutions [71].

Many scholars have adopted the cooperation and RS strategy to optimize the multicenter logistics network [11, 17, 24]. Zhang et al. [15] believed that in the collaborative e-commerce truck carrier, participants can share transportation resources and customer demands to maximize the total profit of the whole alliance and improve vehicle utilization. Wang et al. [14] adopted the cooperation strategy to optimize the MDVRPPDTW with minimization of the operating cost of the transportation network and the total number of vehicles. Deng et al. [21] allowed the capabilities of the logistics facility, the vehicle resources, and the customer information to be shared through RS strategies, and they proved that this strategy can improve the utilization of logistics resources. In the study of Nourinejad et al. [72], vehicles can be used multiple times to reduce fleet size by extending vehicle reservation time. Li et al. [73] adopted the resource-sharing strategy to significantly optimize the logistics network and maximized the utilization of resources. Therefore, the adoption of RS strategy not only helps optimize the logistics costs but also improves the utilization of resources to protect the environment [22, 74].

In summary, the existing literature has provided rich reference materials about MDVRPPD, MDVRPPDTW, and RS, including model formulations and solution algorithms. However, the existing literature related to MDPDPRS has the following limitations. (1) Few studies on MDPDPRS have considered RS, MDVRPPD, and MDVRPPDTW. (2) The fact that a vehicle can be used multiple times on a working day is insufficiently considered in the proposed mathematical models. (3) Most of the proposed solution algorithms in the existing literature only address how to solve MDVRP, MDVRPPD, and MDVRPPDTW, and RS has not been incorporated into the designed algorithms. (4) Most of the existing literature focuses on raising problems and designing algorithms but neglects testing the proposed methods with practical cases.

In consideration of the aforementioned shortcomings, the main contributions of this study to MDPDPRS are as follows: (1) Characteristics of RS, MDVRPPD, and MDVRPPDTW are comprehensively incorporated to enrich the research on MDPDPRS. (2) On the basis of RS, this study proposes and tests that vehicles can be used multiple times and that customer information can be shared to save the transportation resources of logistics networks, which are considered in the proposed mathematical model. (3) A two-stage algorithm is designed to combine RS with vehicle routing optimization to optimize the logistics network. (4) Benchmark and real-world cases are utilized to test and verify the performance and applicability of the proposed model and solution algorithm in this study.

3. Problem Statement

RS is an effective strategy that can optimize logistics operation costs and resource allocation in a multicenter logistics network with pickups and deliveries [21, 35]. In this study, the logistics network consists of multiple distribution centers (DCs), multiple pickup centers (PCs), and multiple customers. The logistics network before and after optimization with RS, which is composed of DC1, DC2, DC3, PC1, PC2, and 52 customers (marked C1, C2, …, C52), is shown in Figure 1. The numerical number near the line represents the time distance between two elements (including facilities and customers).

(a)

(b)

Figure 1: Logistics network optimization diagram of MDPDPRS. (a) Initial logistics network before resource sharing. (b) Optimized logistics network with resource sharing.

In Figure 1(a), the unreasonable arrangement of vehicle routes and the nonsharing of resources are the main reasons for the higher operating costs of the logistics network. First, staggered driving and long-distance service are the two most significant unreasonable arrangements, and they cause additional travel costs. Second, vehicles that violate customers’ time windows often occur. Arriving early and arriving late are generating penalty costs. Finally, the nonsharing of resources between facilities results in the capacity of facilities and transportation resources being left unused. For example, the vehicle no longer works after returning from C2 to DC1 in the service route DC1 ⟶ C25 ⟶ C6 ⟶ C1 ⟶ C2 ⟶ DC1.

In Figure 1(b), the logistics network is optimized through the sharing of customers and transportation resources among facilities. First, the reallocation of customers between facilities facilitates the rearrangement of vehicles, which can avoid staggering and long-distance service. Second, vehicle resources can be better configured through sharing among multiple facilities. For example, V1 is used by DC3 and DC1. V2 and V3 have also implemented distribution and pickup services. The repeated use of vehicles prevents resources from being idle. Finally, the situation of vehicle violating the customers’ time window is avoided.

To clearly demonstrate the discounts that RS brings to logistics network optimization, six indicators are counted, namely, total travel cost (TTC), total penalty cost (TPC), total maintenance cost (TMC), total fixed cost (TFC), number of vehicles (NV), and total operating cost (TOC), which are listed in Table 1. The unit time travel cost using a vehicle can be set to $10/h, and the unit time penalty cost for waiting or late can be set to $20/h, and the maintenance cost using a vehicle in a working period can be defined as $100, and the fixed cost of a facility being used in a working period can be set to $200.

Table 1: Comparison of MDPDPRS before and after optimization.

TTC ($)
TPC ($)
TMC ($)
TFC ($)
NV
TOC ($)
Gap

NV
TOC ($)

Initial
1500
440
1400
1000
14
4340
7
1880

Optimized
760
0
700
1000
7
2460

In Table 1, the gap of TOC and NV in the initial and optimized logistics network is $1880 and 7, respectively, which are highly significant. Before optimization, TTC ($1500), TPC ($440), and TMC ($1400) are the main elements that cause additional logistics costs. In the optimized logistics network, the penalty costs are avoided, and the reduction in the number of vehicles also reduces the maintenance costs with RS. In addition, the travel costs of vehicles are also greatly reduced. Therefore, RS is an effective strategy for optimizing the logistics network of the MDPDPRS.

4. Mathematical Model for MDPDPRS

4.1. Assumptions and Notations

Some necessary assumptions can ensure the usability of the mathematical model [75] and those assumptions are listed as follows [14, 30, 36, 42, 49, 50].

Assumption 1. In a working period, the demand and coordination of customers are stable and known.

Assumption 2. Centralized transportation among logistics facilities is performed by trucks.

Assumption 3. The vehicles can be used multiple times by the same or different logistics facilities.
The relevant notations and their descriptions are listed in Table 2.

Table 2: Notations and description.

Set
Description

SD
Set of DCs, the total number of DCs is ND

SP
Set of PCs, the total number of PCs is NP

SF
Set of DCs and PCs, SF=SP ∪ SD

CD
Set of delivery customers

CP
Set of pickup customers

CU
Set of all customers, U=CD∪CP

V
Set of vehicles, the total number of vehicles is NV

B
Set of trucks, the total number of trucks is NB

Parameters

QcD
Delivery demand of customer c, c ∈ CD

QcP
Pickup demand of customer c, c ∈ CP

[Ec,Lc]
Service time window of customer or logistics facility c, c ∈ CU∪SF

dch
Travel distance between customer or facility c and customer h. c, h ∈ CU∪SF

tch
Travel time of a vehicle driving from customer or facility c to customer h. c, h ∈ CU∪SF

PE
Waiting penalty coefficient of arriving earliness (unit: dollars/unit time)

PL
Tardiness penalty coefficient of arriving delay (unit: dollars/unit time)

W
Number of working periods in a year

T
Number of working days in a working period

Maximum travel time of vehicle v, v ∈ V

Tb
Maximum travel time of truck b, b ∈ B

Fuel consumption rate of vehicle per km (unit: gallon/miles)

fb
Fuel consumption rate of truck b per km (unit: gallon/miles)

Gasoline price (unit: dollars/gallon)

Pb
Gasoline price (unit: dollars/gallon)

Capacity of vehicle v, v ∈ V

Cb
Capacity of truck b, b ∈ B

Cf
Capacity of facility f, f ∈ SF

Annual maintenance cost of vehicle v, v ∈ V

Mb
Annual maintenance cost of truck b, b ∈ B

If
Annual fixed cost of facility f, f ∈ SF

The total service route number of vehicle ,  ∈ V

Gvfk
The departure time of vehicle from facility f in the kth service route. ∈ V, f ∈ SF, k ∈ 

Avck
The arriving time of vehicle at customer or facility c in the kth service route,  ∈ V, c ∈ CU∪SF, k ∈ 

Qof
The number of transported goods from facility o to f, o, f ∈ SF

Variables

xvcfk
 = 1 if customer c is served by vehicle departing from DC or PC f in the kth service route of vehicle ; otherwise, xvcfk = 0,  ∈ V, c ∈ CU, f ∈ SF,k ∈ 

xcof
xcod = 1 if the facility providing logistics service for customer c is changed from facility o to f; otherwise, xcod = 0, c ∈ CU,o, f ∈ SF

xvch
 = 1 if vehicle travels directly from facility or customer c to h; otherwise, xvch = 0,  ∈ V, c,h ∈ SF∪CU.

xbof
xbof = 1, if the goods transported from facility o to f is carried by truck k; otherwise, xbof = 0, b ∈ B, o, f ∈ SF

xbowf
xbowf = 1, if truck k departs from facility o and travels directly from facility to f; otherwise, xbowf = 0, b ∈ B, o,, f ∈ SF

zf
zf = 1 if facility f agrees to share resource, otherwise, zf = 0, f ∈ SF.

4.2. Mathematical Model

A bi-objective mathematical model that minimizes the logistics operational costs (Equation (1)) and the number of vehicles (Equation (2)) is formed to solve the MDPDPRS. In Equation (1), the logistics operation cost is composed of four parts, which are marked by the TTC (Equation (3)), TPC (Equation (4)), TMC (Equation (5)), and TFC (Equation (6)). In Equation (2), represents the shared times of vehicle .

Subject to

Constraint (7) ensures that each customer is served once. Constraint (8) counts the shared times of vehicle . Constraints (9)–(11) ensure that flow conservation is achieved on each customer. Constraints (12) and (13) are the flow balance constraints of the truck. Constraints (14)–(16) ensure that the loading quantity of each vehicle and each truck cannot exceed their capabilities. Constraints (17) and (18) count the quantity of transshipment goods between logistics facilities. Constraints (19) and (20) guarantee that the total service quantity of each facility does not exceed its capacity. Constraints (21) and (22) require that the departure time and return time of each vehicle must meet the service time window of its served facility. Constraints (23) and (24) ensure that each vehicle must provide services for customers within the customers’ service time window. Constraint (25) requires that the total working time of each vehicle does not exceed its maximum working time. Constraints (26)–(28) are used to eliminate the sub-tours of each vehicle and truck. The constraints of relevant binary variables are listed in Constraints (29)–(34).

5. Solution Methodology for MDPDPRS

MDVRPTW and MDVRPPDTW are typical NP-hard problems [12, 18, 41]. Multi-objective optimization algorithm and two-stage algorithm are often designed in combination to solve MDVRPPDTW [14, 23, 36]. Here, a two-stage algorithm with customer clustering first and then vehicle routing optimization is designed to solve MDPDPRS. This two-stage hybrid algorithm is composed of k-means, CW, and NSGA-II algorithms, and named KCW-NSGA-II. In the first stage of KCW-NSGA-II, customers and resources are reconstructed by the k-means clustering algorithm [57, 58]. The main purpose of the second stage is to optimize vehicle routes and find the Pareto optimal solution. The CW algorithm is adopted to construct the initial solution for NSGA-II [33, 37, 64, 65, 67].

The designed algorithm flow is illustrated in Figure 2. Here, Gen is the current number of iterations; MaxGen is the maximum number of iterations; r is the number of iterations of the current internal re-optimization mechanism, which is between clustering and vehicle routing optimization; and MaxR is its maximum number of iterations.

Figure 2: Algorithm flowchart of MDPDPRS.

In Figure 2, the two-stage characteristics of customer clustering first and vehicle routing optimization later are clearly demonstrated. First, a k-means customer clustering mechanism based on Manhattan distance is designed. The clustering results are checked, updated, and saved after finishing the reallocation of all customers. Second, the CW algorithm is adopted to design the initial population and initial feasible solution, which accelerates the speed and possibility of NSGA-II algorithm to find the Pareto optimal solution. Third, the elite strategy and genetic operation of NSGA-II are used to iteratively optimize the generated initial solution to find the Pareto optimal solution, which is mainly embodied in the change of Gen. Fourth, a regulatory re-optimization mechanism between customer clustering and vehicle routing optimization is set up to maintain gene stability during genetic operation, and this mechanism is implemented by the re-updating of r. Finally, if Gen is updated to MaxGen, then the iterative optimization of the algorithm is finished and the found Pareto optimal solution is outputted.

5.1. K-Means Clustering Algorithm

Customer clustering is an important measure to reduce the complexity of solving MDVRPPDTW [36]. K-means algorithm is widely used to solve MDVRPTW due to its simplicity and efficiency [14, 21]. The k-means clustering pseudocode based on Manhattan distance is listed in Algorithm 1.

Input: The datasets, including logistics facility and customer information, such as the coordination, time windows, and demands

 Output: The clustering results

(1)
Step 1: Select k objects as the initial clustering center

(2)
Step 2: Calculate the Manhattan distance between each customer and each clustering center

(3)
Step 3: (Re-)Assign each customer to their closest clustering center

(4)
Step 4: If some customers need to be adjusted among the clustering results, then enter Step 3; otherwise, go to Step 5

(5)
Step 5: Update the clustering centers

(6)
Step 6: Output the clustering results

Algorithm 1: Procedure of k-means algorithm.

5.2. CW Algorithm

A common and effective way to construct the initial solution of VRP is the CW savings algorithm, which is actually a greedy heuristic algorithm [33, 37, 76]. The service time windows of customers and the capacity of vehicles are the main constraints to construct the initial solution [77, 78]. The pseudocode of the CW algorithm designed in this study is listed in Algorithm 2.

 Input: The datasets of customer and facility information including location, time window, demand, and other relevant parameters

 Output: The initial feasible solution

(1)
Step 1: Mark the initial distance saving values with numbers 1, 2, 3, …, (n-1)

(2)
Step 2: Assign vehicles to each customer

(3)
Step 3: Calculate the distance savings of new vehicle routes, which are formed by any two routes

(4)
Step 4: Sort the distance savings in descending order

(5)
Step 5: If new vehicle routes meet the time window and capacity constraints, then go directly to Step 6; otherwise set n=n+1 and return to Step 5

(6)
Step 6: Generate a new vehicle route with two vehicle routes whose distance savings are maximum

(7)
Step 7: Update the distance savings value through the fusion of vehicle routes

(8)
Step 8: Generate new vehicle route with two vehicle routes whose distance savings are maximum

(9)
Step 9: n = 1

(10)
Step 10: If new vehicle routes meet the time window and capacity constraints, then go directly to Step 11; otherwise, return to Step 7

(11)
Step 11: n = n+1

(12)
Step 12: Generate new vehicle route with two vehicle routes whose distance savings are maximum

(13)
Step 13: If there exists a route that serves only one, then go back to Step 7; otherwise enter Step 14

(14)
Step 14: Output the service route of each vehicle

Algorithm 2: Procedure of CW algorithm.

5.3. NSGA-II

NSGA-II is a multi-objective optimization algorithm based on GA, which searches the Pareto optimal solution for multi-objective optimization [38, 64, 68]. Fast nondominated sorting operator, individual crowding distance operator, and elite strategy selection operator are the three key designs of NSGA-II [33, 65]. Here, we suppose that the population is P and n individuals exist, and the individual objective function value of individual i is xi.

5.3.1. Fast Nondominated Sorting Operator

The key design of NSGA-II is to find the Pareto optimal solution. To enhance the possibility of finding the Pareto solution, the fast nondominated sorting operator stratifies the population P according to the quality of individual solutions [33, 64, 65, 67]. This method is a cyclic process of grading based on population fitness. Here, the nondominated solution set and the rank value assigned to the individual are the two indicators for fast nondominated sorting. Assume that Fi represents the nondominated solution set, and ir represents the rank value of individual i. Then, the pseudocode of the fast nondominated sort operator is shown in Algorithm 3.

 Input: Initial population P

 Output: The nondominated sorting results of population P

(1)
Step 1: Setting PP=P, rank=1, F=1

(2)
Step 2: for F=1:n

(3)
Step 3: for i=1:n

(4)
Step 4: for j=1:n && j≠i

(5)
Step 5: Compare the solution quality of xi and xj, and determine the dominant and nondominant relationships of i and j.

        End

(6)
Step 6: If xi is superior to all xj, then the individual i is considered to be a nondominant individual, and the nondominant ranking value of i is rank

        End

(7)
Step 7: The nondominant individuals found in the above steps constitute set SF, which is regarded as the F-level nondominant layer of the population P

(8)
Step 8: PP=PP\SF, ir=ir+1

(9)
Step 9: If PP is an empty set, then enter Step 10, otherwise continuous this cycle

(10)
Step 10: Stratify population P

(11)
Step 11: End

Algorithm 3: Procedure of the fast nondominated sorting operator.

5.3.2. Crowding Distance and Its Comparison Principle

The crowding distance is designed to drive the population to converge to the Pareto optimal solution and maintain the diversity of the population, which is mainly for individuals in the same nondominant layer [38, 64, 68]. We assume that n individuals exist in the nondominant level of SF, and the objective function value of individual i is xi. Then, the crowding distance is calculated as

In Equation (35), represents the sum of the distance between individuals and in each direction of the objective function. Here, the objective function values of individuals need to be sorted before calculating the crowding distance. If the rank values of individuals i and j are ir and jr, then the crowding distance is and , respectively. The individual crowding comparison strategy based on crowding distance and nondominated ranking results is as follows.

(1)If ir > jr, then individual i is the best one. If ir < jr, then individual j is the best one.(2)If ir = jr, then the individual with the most crowded distance is the better one.

5.3.3. Elite Strategy Selection Operator

To prevent the Pareto optimal solution from being lost in the iteration process, an elite strategy selection operator is designed, which selects the optimal solution by nondominated sorting and crowding distance between the parent and offspring populations. Suppose that the current iteration is t and the parent population is Pt, the offspring population is Qt, and Rt is composed by Pt and Qt. First, a fast nondominated sorting is performed for Rt, and the crowding distance is then calculated. On the basis of the crowding distance and the nondominated layer, N individuals with high-quality solution is selected to form a new population Pt+1.

6. Empirical Analyses

6.1. Algorithm Comparison

The standard NSGA-II, GA-PSO, and MOPSO are adopted for comparison to verify the applicability and effectiveness of the proposed KCW-NSGA-II algorithm in solving MDPDPRS [38, 64–66, 68, 79]. The benchmark dataset C-mdvrptw (consisting of instances of 20 groups) is utilized for the test, which is mainly obtained from the website. Networking and emerging optimization and their related characteristics are listed in Table 3. To meet the characteristics of the research object in this study, depots are regarded as logistics facilities, and customers are divided into two types, that is, those with distribution demands and those with pickup demands.

Table 3: Characteristics of benchmark datasets.

Instance
Datasets
Maximum loading capacity
Number of depots
Number of customers

DCs
PCs
Delivery demands
Pickup demands

1
C-mdvrptw-pr01
200
2
2
24
24

2
C-mdvrptw-pr02
195
2
2
48
48

3
C-mdvrptw-pr03
190
2
2
72
72

4
C-mdvrptw-pr04
185
2
2
96
96

5
C-mdvrptw-pr05
180
2
2
120
120

6
C-mdvrptw-pr06
175
2
2
144
144

7
C-mdvrptw-pr07
200
3
3
36
36

8
C-mdvrptw-pr08
190
3
3
72
72

9
C-mdvrptw-pr09
180
3
3
108
108

10
C-mdvrptw-pr10
170
3
3
144
144

11
C-mdvrptw-pr11
200
2
2
24
24

12
C-mdvrptw-pr12
195
2
2
48
48

13
C-mdvrptw-pr13
190
2
2
72
72

14
C-mdvrptw-pr14
185
2
2
96
96

15
C-mdvrptw-pr15
180
2
2
120
120

16
C-mdvrptw-pr16
175
2
2
144
144

17
C-mdvrptw-pr17
200
3
3
36
36

18
C-mdvrptw-pr18
190
3
3
72
72

19
C-mdvrptw-pr19
180
3
3
108
108

20
C-mdvrptw-pr20
170
3
3
144
144

In Table 3, the number of customers and logistics facilities in each instance is different. The first instance comprises four facilities and 48 customers, whereas Instance 10 includes six facilities and 288 customers. In addition, the loading capacity of vehicle used in each instance is differentiated.

Relevant parameters are properly unified to mitigate their effects on algorithm performance. These parameters are set as follows [38, 64–66, 68, 79]: (1) parameters about GA: population size popsize = 200, selection possibility sp = 0.6, crossover possibility cp = 0.9, mutation possibility mp = 0.2; (2) parameters about PSO: popsize = 200, inertia weight iw = 0.85, the personal learning confidences pc = 2, and social learning confidence gc = 3; and (3) other relevant parameters: maximum number of generation genmax = 1200 and velocity of vehicle  = 5. The costs, number of vehicles (NV), and computation time (CT) are calculated to verify the performance of the algorithms and the numerical results are listed in Table 4.

Table 4: Comparison results of the four algorithms.

Instance
KCW-NSGA-II
NSGA-II
GA-PSO
MOPSO

Cost
NV
CT
Cost
NV
CT
Cost
NV
CT
Cost
NV
CT

1
1033
4
71
1178
4
77
1298
5
82
1593
5
98

2
1787
8
72
1724
9
91
2172
10
93
2115
8
119

3
2742
17
110
3121
18
133
2879
18
103
3309
18
127

4
3216
20
129
4027
21
146
3371
21
171
3579
21
184

5
3693
18
181
3827
22
239
3835
22
217
4121
30
264

6
3569
15
306
3671
23
290
4008
24
301
4577
33
298

7
1687
7
85
2386
9
83
1767
8
97
2044
14
79

8
2093
12
95
2394
13
141
2181
12
114
2361
12
147

9
2614
19
218
4010
22
210
2653
20
203
3167
30
240

10
3123
24
258
3355
24
311
3277
25
277
3459
29
325

11
1118
4
77
1405
5
74
1190
6
85
1754
11
101

12
2068
10
94
2692
11
96
2394
11
88
2135
11
114

13
2570
18
97
2655
18
103
2690
22
119
3282
28
99

14
3028
21
162
3564
24
186
3108
23
162
3620
25
172

15
3709
22
209
4117
26
240
3812
27
241
4272
33
248

16
3993
25
262
4072
25
305
4154
28
301
4366
34
339

17
1578
7
69
1763
7
87
1665
7
65
1652
7
126

18
2324
12
66
2428
12
110
2430
14
109
2635
19
166

19
2440
20
212
2790
21
206
2493
22
202
3029
22
241

20
3295
23
186
3692
25
316
3546
25
213
3421
26
309

Average
2584
15
148
2944
17
172
2746
18
162
3025
21
190

t-test

−13.95
−3.73
−3.29
−6.44
−4.76
−3.41
−8.60
−4.84
−5.31

p-value

9.8E-12
7.2E-04
1.9E-03
1.8E-06
6.8E-05
1.5E-03
2.8E-08
5.7E-05
2.0E-05

The numerical results shown in Table 4 demonstrate that the proposed algorithm KCW-NSGA-II performs better than the other three algorithms. First, the average cost of the four algorithms is $2584, $2944, $2746, and $3025, respectively. By contrast, the costs and the number of vehicles optimized by KCW-NSGA-II are the most economical solution compared with the other three algorithms in each instance. Second, the value of t-test also shows that the KCW-NSGA-II is significantly different from the other three algorithms. In addition, the proposed algorithm can obtain the optimal solution quickly. Therefore, the proposed algorithm KCW-NSGA-II outperforms the other three algorithms. Moreover, this algorithm can be adjusted to address problems such as VRPMDP, VRPSDP, and PVRP.

6.2. Data Source and Relevant Parameter Setting

As an inland international logistics hub and an open highland, Chongqing is a new first-tier city in China. Therefore, as our numerical experiments, the logistics network adopted from Chongqing is appropriate to verify the applicability of this study. Six logistics facilities (i.e., DC1, DC2, DC3, PC1, PC2, and PC3) and 220 customers are the main elements of this real-world logistics network. The information and characteristics of these elements are listed in Table 5, and the spatial allocation information is plotted in Figure 3.

Table 5: Characteristics of the logistics network.

Symbol
Description
Number of served customers
Mark of customers

DC1 and its customers
27
C1 - C27

DC2 and its customers
36
C28 - C63

DC3 and its customers
39
C64 - C102

PC1 and its customers
46
C103 - C148

PC2 and its customers
31
C149 - C179

PC3 and its customers
41
C180 - C220

Figure 3: Spatial distribution of DCs, PCs, and their customers.

In Table 5, the number of customers served by the six logistics facilities is 27, 36, 39, 46, 31, and 41, respectively. In Figure 3, an obvious feature is that the customer allocation of each facility is relatively dispersed. The service area edge of each facility is not a clear division. In Table 6, the initial vehicle routes for the logistics network are shown, including the specific information of each service route.

Table 6: Distribution routes of DCs and pickup routes of PCs in the initial logistics network.

Facility
Vehicle
Departure time
Arriving time
Route

DC1
V1
162
689
DC1⟶C19⟶C1⟶C17⟶DC1

V2
0
906
DC1⟶C13⟶C7⟶C20⟶C12⟶DC1

V3
159
619
DC1⟶C8⟶C25⟶C11⟶C3⟶C4⟶C10⟶DC1

V4
33
730
DC1⟶C27⟶C24⟶C21⟶C9⟶C26⟶C14⟶C15⟶C18⟶DC1

V5
0
924
DC1⟶C23⟶C22⟶C6⟶C16⟶C2⟶C5⟶DC1

DC2
V6
0
1037
DC2⟶C42⟶C45⟶C48⟶C34⟶C35⟶C50⟶C54⟶DC2

V7
30
1016
DC2⟶C58⟶C39⟶C41⟶C47⟶C37⟶C30⟶C43⟶C40⟶DC2

V8
0
1014
DC2⟶C46⟶C61⟶C28⟶C59⟶C36⟶C32⟶C62⟶C52⟶C60⟶DC2

V9
0
319
DC2⟶C63⟶C51⟶C49⟶DC2

V10
35
823
DC2⟶C56⟶C55⟶C38⟶C53⟶DC2

V11
10
1196
DC2⟶C31⟶C33⟶C44⟶C29⟶C57⟶DC2

DC3
V12
121
1002
DC3⟶C83⟶C98⟶C95⟶C77⟶C73⟶C93⟶C75⟶DC3

V13
92
831
DC3⟶C79⟶C89⟶C92⟶C69⟶C101⟶C90⟶C68⟶DC3

V14
109
1176
DC3⟶C86⟶C99⟶C78⟶C82⟶C100⟶C64⟶DC3

V15
0
865
DC3⟶C102⟶C81⟶C85⟶C76⟶C70⟶C80⟶C97⟶C88⟶DC3

V16
0
936
DC3⟶C74⟶C67⟶C84⟶C87⟶C71⟶C72⟶C91⟶C65⟶C96⟶DC3

V17
0
295
DC3⟶C94⟶C66⟶DC3

PC1
V18
0
353
PC1⟶C114⟶C145⟶C119⟶PC1

V19
19
891
PC1⟶C117⟶C125⟶C121⟶C120⟶C130⟶C113⟶C128⟶C111⟶C118⟶PC1

V20
4
992
PC1⟶C115⟶C144⟶C116⟶C123⟶C140⟶C122⟶C106⟶C139⟶C124⟶PC1

V21
95
1103
PC1⟶C135⟶C104⟶C136⟶C103⟶C147⟶C142⟶C141⟶C143⟶C112⟶PC1

V22
138
951
PC1⟶C148⟶C133⟶C146⟶C107⟶C131⟶C137⟶C138⟶PC1

V23
0
1055
PC1⟶C134⟶C132⟶C105⟶C127⟶C129⟶C126⟶C110⟶C109⟶C108⟶PC1

PC2
V24
0
883
PC2⟶C165⟶C162⟶C153⟶C151⟶C170⟶C168⟶C158⟶PC2

V25
37
1067
PC2⟶C149⟶C179⟶C174⟶C176⟶C169⟶C166⟶C159⟶0

V26
0
1115
PC2⟶C156⟶C172⟶C167⟶C157⟶C155⟶C152⟶C163⟶C150⟶PC2

V27
18
819
PC2⟶C171⟶C178⟶C160⟶C161⟶C164⟶C177⟶C154⟶C173⟶C175⟶PC2

PC3
V28
202
1031
PC3⟶C192⟶C201⟶C182⟶C184⟶PC3

V29
127
918
PC3⟶C220⟶C183⟶C193⟶C214⟶C207⟶C217⟶C205⟶C202⟶C180⟶PC3

V30
0
720
PC3⟶C206⟶C211⟶C209⟶C212⟶C200⟶C215⟶C196⟶PC3

V31
23
1121
PC3⟶C191⟶C185⟶C187⟶C188⟶C216⟶C199⟶PC3

V32
217
674
PC3⟶C181⟶C213⟶C219⟶C203⟶C195⟶C210⟶PC3

V33
66
918
PC3⟶C204⟶C198⟶C194⟶C197⟶C186⟶C208⟶C218⟶C190⟶C189⟶PC3

In Table 6, the total number of vehicles used in the initial logistics network is 33, and the number of vehicles used at each facility is 5, 6, 6, 6, 4, and 6, respectively. In addition, some vehicles return to their origin early, such as V9, V10, V17, and V18, indicating that these vehicle resources are underutilized. The service vehicle routes of DC1 and the 27 customers it serves in the logistics network are shown in Figure 4.

Figure 4: Initial vehicle service routes of DC1.

In Figure 4, the service routes of V3, V4, and V5 are relatively complex. V5 performs delivery services for customers C23, C22, C6, C16, C2, and C5. However, these customers may be closer to DC3 on the basis of the perspective of spatial distribution. Therefore, optimizing this logistics network is necessary. The values of the relevant parameters used in this real-world case study are shown in Table 7 [38, 64, 65, 68].

Table 7: Relevant parameter setting.

Parameter
Numerical value
Parameter
Numerical value

PE
$35 per hour
Cb
1000

PL
$35 per hour
Cf
2000

W
52

$13000

T
7
Mb
$20000

10 hour
[IDC1, IDC2, IDC3, IPC1, IPC2, IPC3]
[$315, $427, $533, $578, $612, $590]

Tb
15 hour
Population size
300

0.363 gallon per miles
Maximum generations
1000

fb
0.566 gallon per miles
Crossover probability
0.8

$6.18 per gallon
Mutation probability
0.2

Pb
$6.18 per gallon
Travel speed of vehicle
40

200
Travel speed of truck
60

6.3. Optimization Results

Clustering customers to optimize resource allocation is the first step in optimizing the logistics network. The customer clustering results of this logistics network by k-means algorithm are shown in Figure 5.

(a)

(b)

Figure 5: Customer clustering results with k-means algorithm. (a) Customer clustering results with DCs as cluster centers. (b) Customer clustering results with PCs as cluster centers.

In Figure 5, the service relationship between customers and facilities is optimized by clustering. Each customer is covered by the logistics facility that is located close to that customer. On the whole, the service area of each logistics facility has been obviously allocated. For example, C23, C22, C6, C16, and C2 are served by DC1 before clustering; however, they are also served by DC3. Statistical analysis of customers whose service relationship has changed like those five customers is the key to handle centralized transportation. The details of the amount of goods transferred among facilities are shown in Figure 6.

Figure 6: Description of transmissions and distances among facilities.

In Figure 6, the amount of goods shipped from DC1 to DC2 is zero; thus, the initial customers of DC1 are not allocated to DC2 with clustering. The amount of goods transported from DC2 to DC1 is 170, which indicates that the service relationship of some customers has changed from DC2 to DC1. Here, the spatial distance between DC1 and DC2 is 33.07 km. The centralized transportation service routes of the truck are listed in Table 8.

Table 8: Optimized truck routes of centralized transportation.

Truck
Service routes

T1
DC1⟶DC2⟶DC1

DC2⟶DC3⟶DC1⟶DC2

DC3⟶DC2⟶DC1⟶DC3

T2
PC1⟶PC2⟶PC3⟶PC1

PC2⟶PC3⟶PC1⟶PC2

PC3⟶PC2>PC1⟶PC3

In Table 8, T1 and T2 serve three routes. At the end of each service route, the trucks should return to their origin. The activity of T1 occurs early on each workday, and that of T2 occurs late on each workday. Table 9 shows the optimized vehicle service routes.

Table 9: Vehicle distribution and pickup sharing routes in the optimized logistics network.

Vehicle
Facility
Departure time
Arriving time
Origin
Destination
Vehicle distribution and pickup sharing routes

V1
DC1
200
602
DC1
DC3
C86⟶C13⟶C20⟶C11⟶C4⟶C12⟶C7⟶C5⟶C1⟶C25⟶C3⟶C8

V2
107
427
DC1
DC1
C24⟶C79⟶C39⟶C27⟶C89⟶C74⟶C51⟶C14⟶C9⟶C92

V2
577
1102
DC1
DC1
C17⟶C90⟶C101⟶C78⟶C30⟶C18⟶C65⟶C35⟶C96⟶C75⟶C50⟶C54⟶C57

V3
0
773
DC1
DC1
C63⟶C21⟶C45⟶C19⟶C26⟶C69⟶C37⟶C15⟶C10⟶C68

V4
0
420
DC2
DC2
C46⟶C31⟶C102⟶C61⟶C33⟶C56⟶C85⟶C66⟶C81⟶C76⟶C32

V4
DC2
491
1007
DC2
DC2
C62⟶C52⟶C80⟶C70⟶C99⟶C88⟶C53⟶C100⟶C43⟶C64⟶C60

V5
65
526
DC2
DC2
C49⟶C58⟶C94⟶C28⟶C36⟶C59⟶C41⟶C47⟶C38⟶C44

V1
672
938
DC3
DC3
C73⟶C97⟶C42⟶C2⟶C82⟶C29⟶C16⟶C40

V6
DC3
121
433
DC3
DC3
C83⟶C22⟶C6⟶C84⟶C67⟶C48⟶C87⟶C55⟶C23

V6
450
712
DC3
DC3
C98⟶C95⟶C77⟶C34⟶C71⟶C72⟶C93⟶C91

V7
69
556
PC1
PC1
C119⟶C115⟶C198⟶C130⟶C194⟶C161⟶C186⟶C128⟶C116

V7
PC1
640
1089
PC1
PC1
C122⟶C134⟶C140⟶C131⟶C139⟶C137⟶C118⟶C138⟶C109⟶C218⟶C110⟶C108⟶C124

V8
91
351
PC1
PC1
C145⟶C135⟶C104⟶C172⟶C126⟶C144⟶C204⟶C125

V8
393
906
PC1
PC1
C113⟶C148⟶C146⟶C133⟶C152⟶C107⟶C200⟶C215⟶C208⟶C106⟶C202

V9
37
571
PC2
PC2
C219⟶C220⟶C103⟶C213⟶C206⟶C174⟶C192⟶C169⟶C173⟶C175⟶C184

V9
PC2
592
1104
PC2
PC2
C149⟶C165⟶C162⟶C179⟶C196⟶C207⟶C212⟶C195⟶C189⟶C203⟶C201⟶C182⟶C166⟶C159

V10
0
630
PC2
PC3
C114⟶C211⟶C156⟶C120⟶C176⟶C209⟶C197⟶C177⟶C164⟶C154⟶C127⟶C129⟶C111

V10
692
997
PC3
PC3
C141⟶C188⟶C143⟶C163⟶C150⟶C216⟶C190⟶C112

V11
24
306
PC3
PC3
C117⟶C191⟶C180⟶C132⟶C160⟶C171⟶C183⟶C155

V11
PC3
355
843
PC3
PC3
C167⟶C136⟶C153⟶C142⟶C147⟶C187⟶C185⟶C123⟶C205⟶C170

V12
49
170
PC3
PC3
C121⟶C105⟶C178⟶C217

V12
192
928
PC3
PC3
C181⟶C157⟶C193⟶C151⟶C214⟶C210⟶C168⟶C158⟶C199

In Table 9, the number of vehicles used jointly by the six logistics facilities is 12. Some of the vehicles are used multiple times within and between the facilities. For example, V1 performs the route DC1 ⟶ C86 ⟶ C13 ⟶ C20 ⟶ C11 ⟶ C4 ⟶ C12 ⟶ C7 ⟶ C5 ⟶ C1 ⟶ C25 ⟶ C3 ⟶ C8 ⟶ DC1 and the route DC3 ⟶ C73 ⟶ C97 ⟶ C42 ⟶ C2 ⟶ C82 ⟶ C29 ⟶ C16 ⟶ C40 ⟶ DC3, which occur in DC1 and DC3, respectively. V10 provides service for PC2 and PC3 successively. V2, V4, and V6 are shared in DC1, DC2, and DC3, respectively.

To clarify the effect of the proposed model and algorithm, the gap of the cost and the number of vehicles in the initial and optimized logistics network are counted and listed in Table 10. Here, the TOC of the logistics facility includes the TTC, TFC, TPC, and TMC. Centralized transportation is a special project generated by the sharing of customer information and transportation resources among facilities. Therefore, the costs of centralized transportation should be jointly borne by all the facilities participating in the sharing. Similarly, given that the vehicle is shared and some vehicles are used multiple times, the maintenance of the vehicles should be co-paid by all facilities.

Table 10: Comparison before and after optimization.

Facility
Initial

Optimized
Gap

TOC
NV
TOC
NV
TOC
NV

TTC
TFC
TPC
TMC

TTC
TFC
TPC
TMC

DC1
2210
315
27
179
5
2047
315
0
429
12
8859
19

DC2
3581
427
55
214
6
1738
427
0

DC3
3730
533
44
214
6
1134
533
0

PC1
4900
578
65
214
6
2064
578
0

PC2
3067
612
22
143
4
2025
612
0

PC3
3506
590
33
214
6
2361
590
0

Centralized transportation
−
−
−
−
−
1651
−
0
110
2

Total
25473
33
16614
14
−
−

In Table 10, the fixed costs of each facility are stable, which are $315, $427, $533, $578, $612, and $590, respectively. Reducing the travel, penalty, and maintenance costs is the main objective of optimizing the logistics operating cost. The TOC of the initial logistics network is $25473, whereas the optimized TOC is $16614, which indicates that the logistics network is significantly improved. Transportation resources are greatly saved, as the number of vehicles before and after optimization is 33 and 14, respectively. The gap of TTC before and after optimization is shown in Figure 7, which can directly prove the optimization effect of vehicle routes.

Figure 7: Comparison results of TTC before and after optimization.

In Figure 7, the initial TTC of DC1, DC2, DC3, PC1, PC2, and PC3 is $2210, $3581, $3730, $4900, $3067, and $3506, respectively. The optimized TTC of each facility is $2047, $1738, $1134, $2064, $2025, and $2361, respectively. The difference of TTC before and after optimization is relatively obvious. Although the transportation cost generated by the centralized transportation is $1651, the logistics network is still significantly optimized on the whole.

6.4. Analysis and Discussion

The optimization of the logistics network is divided into four cases based on the RS strategy to analyze the influence of RS on the optimization of the logistics network. In Case 1, customer information and transportation resources are privately owned by each logistics facility. In Case 2, RS is adopted by DCs (i.e., DC1, DC2, and DC3). Correspondingly, in Case 3, the members participating in RS are PCs (i.e., PC1, PC2, and PC3). In Case 4, all six facilities receive the RS strategy. Transportation resources can be used jointly by DCs and PCs in Case 4. Table 11 and Figure 8 show the numerical optimization results of the logistics network in the four scenarios.

Table 11: Comparison results of the four cases.

Cases
TTC ($)
TPC ($)
TFC ($)
NV
TMC ($)
TOC ($)
Gap

NV
TTC ($)
TOC ($)

Case 1
20994
246
3055
33
1178
25473

Case 2
17038
120
3055
23
841
21053
10
3956
4420

Case 3
16977
126
3055
24
876
21034
9
4017
4439

Case 4
13020
0
3055
14
539
16614
19
7974
8859

Figure 8: Comparison results of TTC, TC, TMC, and NV in the four cases.

In Table 11 and Figure 8, Case 4 outperforms Cases 2 and 3 in most aspects. First, the TOC of Cases 2 and 3 is $17038 and $16977, respectively, which are larger than $13020 in Case 4. Second, the number of vehicles and the TMCs of Scenario 4 is 14 and $539, respectively, which indicates that Case 4 is more conducive to saving resources. Finally, the gap of TOC between Cases 4 and 1 is $8859. Therefore, if the six logistics facilities adopt RS simultaneously, then the logistics costs can be optimized better and the transportation resources can be saved considerably.

6.5. Management Insights

In this study, the RS strategy optimizes the logistics network significantly by reallocating customers, including the logistics operation costs and the number of vehicles. Therefore, the management insights obtained in this study are as follows:

(1)In a multi-depot, large-scale logistics network with pickups and deliveries, customer information, facility capacity, and transportation resources can be shared to amplify resource utilization by introducing RS strategies. RS is not only conducive to the operation of logistics facilities, including the use of logistics costs and transport resources, but also conducive for providing customers with more convenient logistics services. In a logistics network with vehicle sharing, vehicles are used multiple times within and between facilities to avoid idle vehicle resources. In addition, the sharing of customer information helps facilities provide logistics services to customers with higher quality. Therefore, participating actively in RS is remarkably necessary for LSPs.(2)From the perspective of local traffic management departments and environmental departments, the optimization of the logistics network through RS not only reduces transportation resources but also relieves the local traffic pressure. In addition, the efficient use of resources promotes the development of green environment. Therefore, local government departments can actively support and give incentive policies to promote RS among local logistics facilities to optimize the logistics network and ease the traffic environment. Moreover, the effective sharing mechanism in the multicenter logistics network with pickups and deliveries can enhance the efficient operation of the logistics system and promote the green and sustainable development of the local intelligent logistics system. Therefore, the introduction of RS can promote the construction and development of local smart and green urban logistics with the incentive of departments and the active participation of facilities.

7. Conclusions

This study handles the MDPDPRS, which optimizes the logistics network by sharing customer and transportation resources. The reconfiguration of resources and customers improve the operating efficiency of the logistics network. The main contributions of this study include the following aspects. First, the MDPDPRS is modeled as a bi-objective mathematical model to optimize the total logistics operating cost and number of vehicles. Second, a two-stage hybrid algorithm is designed to solve the MDPDPRS, which contains the k-means, CW, and NSGA-II algorithms. Third, the application of the proposed mathematical model and methodology are improved by the numerical result of a real-world case study and benchmark.

In view of the shared transportation resources and customers’ information, a two-stage algorithm is designed, which initially clusters customers to reconfigure the resources and then optimizes the vehicle routes. On the one hand, k-means algorithm, which clusters customers based on Manhattan distance, helps reduce the difficulty of solving MDPDPRS and enables vehicle resources to be used in a centralized manner multiple times. On the other hand, the combination of CW and NSGA-II algorithms improves the global searching capability and the speed of the algorithm in finding the Pareto optimal solutions.

The numerical results of a real-word case study, which is obtained in Chongqing, China, is discussed and analyzed to improve the application and performance of the designed mathematical model and methodology in solving practical problems similar to the MDPDPRS. The gap of the logistics operating costs and number of vehicles before and after the optimization are $8859 and 19, respectively, which verify the effectiveness of the model and methodology proposed in this study. In addition, the algorithm comparison results of benchmarks (from C-mdvrptw datasets) verify that the proposed KCW-NSGA-II algorithm is superior to the standard NSGA-II, GA-PSO, and MOPSO. The results of numerical discussion on the four cases in which different RS strategies are adopted prove that RS is helpful to optimize logistics costs and save transportation resources.

In this study, the bi-objective mathematical model and methodology of MDPDPRS are designed, which provide references for the reconfiguration of resources and the optimization of logistics operation costs. In view of the limitations of the current study and the dynamic development of the logistics industry, further research can be considered from the following aspects. (1) The dynamic change in customer demands and customer satisfaction are the two aspects that the realistic LSPs focus on, and these aspects can be added into the study of MDPDPRS. (2) Constructing a dynamic mathematical programming model and designing an exact algorithm to find the exact solution of MDPDPRS are worthy of research. (3) Exploring the approaches to realize RS in a large logistics network and the means to reduce the effect of logistics transportation on the environment can be considered in the study of MDPDPRS. (4) Considering the cost sharing mechanism under the RS mechanism to promote the formation of collaboration and maintain its stability can enrich the study of MDPDPRS.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they do not have any conflicts of interest.

Acknowledgments

This research was supported by National Natural Science Foundation of China (Project Nos. 71871035 and 41977337), the Fundamental Research Funds for the Central Universities (grant no. 22120210009), Humanity and Social Science Youth Foundation of Ministry of Education of China (18YJC630189), Key Science and Technology Research Project of Chongqing Municipal Education Commission (KJZD-K202000702), Key Project of Human Social Science of Chongqing Municipal Education Commission (No. 20SKGH079), Social Science Foundation of Chongqing of China (2019YBGL054), and Chongqing Graduate Tutor Team Construction Project (No. JDDSTD2019008). This research was also supported by 2018 Chongqing Liuchuang Plan Innovation Project (cx2018111).

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