The Vehicle Routing Problem with Simultaneous Pickup and Delivery Considering the Total Number of Collected Goods

Abstract

As a consequence of e-commerce development, large quantities of returned goods are shipped every day. The vehicle routing problem with simultaneous delivery and pickup (VRPSDP) has become one of the most important areas of logistics management. Most related studies are aimed at minimizing travel time. However, the total number of collected goods is also very important to logistics companies. Thus, only considering the traveling time cannot reflect actual practice. To effectively optimize these operations for logistics companies, this paper introduces the vehicle routing problem with simultaneous pickup and delivery considering the total number of collected goods. Based on the principles of considering the number of collected goods, a bi-objective vehicle routing model minimizing the total travel time and maximizing the total number of collected goods simultaneously is developed. A polynomial time approximation algorithm based on the $\epsilon$-constraint method is designed to address this problem, and the approximation ratio of the algorithm is analyzed. Finally, the validity and feasibility of the proposed model and algorithm are verified by test examples, and several managerial insights are derived from the sensitivity analysis.

1. Introduction

The increasingly mature e-commerce has promoted the rapid development of logistics distribution industry, and the scale of reverse logistics caused by return has also increased. According to the statistics of the National Retail Federation of the United States, the return rate of e-commerce reached 20–30%. In December 2019, UPS alone collected about 1 million return packages every day. During China’s double 11 in 2018, as of 13 November, Tmall returned 9.27 billion yuan’s commodities. The high return volume makes the logistics and distribution enterprises to provide distribution and recycling services to customers at the same time. The traditional vehicle routing problem (VRP) cannot describe the characteristics of simultaneous pick-up and delivery in the logistics and distribution process, so an increasing number of scholars have turned to its branch problem-the vehicle routing problem with simultaneous delivery and pickup (VRPSDP).

The VRPSPD refers to the situation where the vehicle starts from the distribution center to deliver goods at the designated demand point and also needs to bring the goods at the demand point back to the distribution center. How to arrange the vehicle route so that the vehicle can complete the pickup service request at the same time of delivery, or the transportation cost is the smallest when all the pickup service requests are completed. It was first proposed by Min in 1989 [1]. The optimization problem of the simultaneous pickup and delivery vehicle routing between a book center and 22 local libraries with the goal of minimizing distribution costs was studied. Its path planning has a great influence on distribution efficiency. Subsequently, this problem has attracted the attention of an increasing number of researchers. About 16.3% of the vehicle routing problem (VRP) research papers published from 2009 to 2017 belonged to the VRPSDP [2].

Almost all the studies aimed at minimizing the travel time and modeling and designing algorithms for the VRPSDP based on known or uncertain pickup demand and single or multiple vehicles. However, in practice, the total number of collected goods is also very important to logistics companies and should be considered in route optimization. In this paper, according to the actual demand and the lack of existing research, under vehicle capacity constraints and considering the travel time and the total number of collected goods at the same time, a bi-objective VRPSDP optimization model is established to minimize the travel time and maximize the total number of collected goods. An approximate algorithm is designed to solve the problem, and an example is given to verify the model and algorithm.

The rest of this study is organized as follows. Section 2 describes a literature review on the VRPSDP with known pickup demand and uncertain pickup demand. Section 3 defines the bi-objective mathematical VRPSDP model, including the description of the problem and the model. The model analysis and solution are introduced in Section 4 and Section 5 present the results of the testing example. Finally, Section 6 provides concluding statements on our work.

2. Literature Review

2.1. The VRPSPD with Known Pickup Demand

The VRPSDP problem was first proposed by Min in 1989 [1]. Some accurate algorithms had been designed to solve this problem, such as the branch-and-price method [3], the branch-and-cut method [4,5,6,7], and a combination of the branch-and-cut-and-price method [8,9].

Because the number of feasible solutions increases factorial with the increase of demand points, there are only a few accurate algorithms to solve the problem. The heuristic algorithm and meta-heuristic algorithm are the main algorithms to solve the simultaneous pickup and delivery vehicle routing problem. Dethloft [10] proposed a heuristic algorithm based on insertion and test instances to solve real-life problems. Kaya et al. developed a hybrid metaheuristic algorithm based on an ant colony system (ACS) and a variable neighborhood search (VNS) for VRPSDP’s solution [11]. Zachariadis solved this problem via a common local-search method for the routing aspects, integrated with a packing heuristic for developing feasible item loadings [12]. Simsir et al. established the mathematic model of VRPSDP with a single distribution center and a single type of vehicle, and constructed the artificial bee swarm algorithm to solve it [13].

Li [14] developed a metaheuristic method based on an iterated local search for the multi-depot vehicle routing problem with simultaneous deliveries and pickups (MDVRPSDP). Hs et al. [15] also discussed the VRPSDP with multiple distribution centers, established a model, and solved it. Avic [16] proposed the heterogeneous vehicle routing problem with a simultaneous pickup and delivery (HVRPSPD) model and developed a hybrid local search algorithm in which a non-monotone threshold adjusting strategy is integrated with tabu search. Zhang [17] addressed a multi-commodity many-to-many vehicle routing problem with simultaneous pickup and delivery (M-M-VRPSPD) for a fast fashion retailer in Singapore. Different from other widely studied pickup and delivery problems, the unique characteristics are collecting products from customers were encouraged to be reallocated to fulfill the demands of other customers. Wang [18] formulated and solved an optimization of multi-depot pickup and delivery problems with resource sharing.

In order to solve the VRPSDP with a time window, Wang [19] proposed a particle swarm optimization algorithm and verified its effectiveness, and the same problem was solved by and Tabu search algorithm [20].

As can be seen above, for the VRPSDP when the demand is known, scholars established the model based on the number of distribution centers, the type of vehicles, time windows et al., mostly with the goal of minimizing the transportation cost. The static optimization method was used to solve the problem. Focus on designing different algorithms to find the approximate optimal solution, trying to find a more stable and efficient algorithm [21].

2.2. The VRPSPD with Uncertain Pickup Demand

Ignoring the uncertainty in the transportation process is the main reason for the decrease in service level and the increase in cost, so some scholars have studied the situation of uncertain pickup demand. Liu et al. [22] constructed a mixed integer programming model for the VRPSDP with a single distribution center and a single type of vehicle, in which the pickup demand follows a random binomial distribution, with the goal of minimizing the expected travel time. Then, they designed a hybrid ant colony system optimization algorithm. Peng [23] established a VRPSDP model of a single distribution center and a single type of vehicle with a normal pickup distribution. Jia et al. simplified the problem to an asymmetric traveling salesman problem with a time window and solved it using a rolling time-domain scheduling algorithm [24]. Hu et al. [25] studied the VRPSDP with incompatible goods and uncertain pickup demand and handled the uncertain pickup demand through two stages: The first stage obtained the pickup demand through a probability distribution, resulting in an a priori path; and the second stage introduced the vehicle loading rate control parameters to judge and solve the service demand points. Based on pre-optimization and re-optimization strategies, Fan et al. [26] constructed a two-stage vehicle routing problem model with uncertain pickup demand. In the pre-optimization stage, the vehicle was assigned to the demand point based on a random chance constraint mechanism and vehicle loading constraint. The pre-optimization scheme was generated. In the re-optimization stage, the route of the failure point and its subsequent demand point was adjusted. Finally, a hybrid variable neighborhood search algorithm was designed to solve the problem. On this basis, the VRPSDP with random demand under the multicenter joint distribution mode of multiple distribution centers was further studied, and a two-stage path optimization model was constructed [27]. According to the characteristics of the problem, an adaptive variable neighborhood cultural gene algorithm was designed to solve the problem. With the goal of minimizing travel costs, scholars have modeled and designed algorithms to solve the situation in which single distribution centers, multi-vehicle, pick-up demand, and delivery demand were all random [28,29].

Zhou [30] studied the optimization model and algorithm of the multi-depot VRPSDP under fuzzy demand. Ma [31] assumed that the pickup demand is a fuzzy random variable, established the VRPSDP model, and designed a genetic algorithm based on fuzzy random simulation to solve it. Combined with credibility theory, Fan et al. [32] modeled the VRPSDP with fuzzy pickup demand and finally transformed it into a deterministic problem to solve.

In summary, it was found that in the VRPSDP with uncertain demands, most of the existing studies randomized or fuzzed the uncertain quantity of pickup, assuming that the quantity of pickup at the demand point obeys a certain distribution (probability distribution or fuzzy distribution). Then find out the minimum expected transportation cost and seek the optimal scheme in the average sense.

Almost all of the mentioned studies aimed at minimizing travel time and modeling and designing algorithms for the VRPSDP. Since the total number of collected goods is very important to logistics companies and should be considered in route optimization. However, no such total number of collected goods-related models has been proposed till now.

3. Problem Description and Modeling

3.1. Problem Description and Symbolic Description

Let $G=(V,E)$ be an undirected complete graph, with vertex set $V=\left\{v_0\right\}\cup V_0$ and edge set $E=\left\{e(v_i,v_j)|v_i,v_j\in V\right\}$, where vertex $v_0$ represents the distribution center and $V_0=\left\{v_1,v_2,\dots v_n\right\}$ represents the demand point set of $G$. The distribution center $v_0$ has $l$ vehicles with a maximum carrying capacity of $Q$, and the vehicles are sufficient to complete the distribution task. $t_{ij}$ is the time when the vehicle passes through edge $e\left(v_i,v_j\right)$, and $d_i$ and $p_i$ are the delivery and pickup quantities at demand point $v_i$, respectively. $Q_{ijk}$ is the loading volume of vehicle $k$ from demand point $v_i$ to $v_j$. Now, we need to arrange vehicles to start from the distribution center $v_0$ to deliver the goods to the designated demand point and determine whether to return goods to $v_0$ according to the situation. The delivery demand and pickup demand of each demand point are known before departure, and the demand to be pickup cannot be split. The problem is how to determine the vehicle route and whether to pick up goods after reaching the demand point while not exceeding the carrying capacity of the vehicle and completing the delivery task so as to make the travel time as small as possible and the total number of goods collected as much as possible.

In order to facilitate the follow-up discussion, the decision variables required for modeling are given first.

$x_{ijk}=$ 1 if vehicle $k$ travels directly from demand point $v_i$ to $v_j$, and 0 otherwise. $y_{ik}=1$ if vehicle $k$ collects goods at demand point $v_i$, and 0 otherwise.

3.2. Modeling

Using the above-defined parameters and variables, we present the formulation of a bi-objective mathematical VRPSDP model.

$$f_1=\max {\displaystyle \sum _{i=1}^n{\displaystyle \sum _{k=1}^l{p}_i}}{y}_{ik}$$

(1)

$$f_2=\min {\displaystyle \sum _{i=0}^n{\displaystyle \sum _{j=0}^n{\displaystyle \sum _{k=1}^l{t}_{ij}}}}{x}_{ijk}$$

(2)

$$\mathrm{subject} \mathrm{to} {\displaystyle \sum _{i=0}^n{\displaystyle \sum _{k=1}^l{x}_{ijk}}}=1,(j=1,2,\dots ,n)$$

(3)

$${\displaystyle \sum _{i=0}^n{x}_{ijk}}={\displaystyle \sum _{i=0}^n{x}_{jik}},(j=1,2,\dots ,n;k=1,2,\dots ,l)$$

(4)

$${\displaystyle \sum _{i=1}^n{x}_{i0k}}={\displaystyle \sum _{j=1}^n{x}_{0jk}},(k=1,2,\dots ,l)$$

(5)

$$Q_{ijk}\le Q,(i=0,1,2,\dots ,n;j=0,1,2,\dots ,n;k=1,2,\dots ,l)$$

(6)

$${\displaystyle \sum _{i=0}^n{Q}_{ijk}}-d_j+p_j{y}_{jk}={\displaystyle \sum _{i=0}^n{Q}_{jik},}(j=1,2,\dots ,n;k=1,2,\dots ,l)$$

(7)

$$x_{ijk}\le y_{ik},(i=1,2,\dots ,n;j=1,2,\dots ,n;k=1,2,\dots ,l)$$

(8)

$$x_{ijk}=\begin{array}{ccc}0& or& 1,(i=\end{array}0,1,\dots ,n;j=0,1,2,\dots ,n;k=1,2,\dots ,l)$$

(9)

$$y_{ik}=0 or 1,(i=1,2,\dots ,n;k=1,2,\dots ,l)$$

(10)

Objective function (1) maximizes the total number of goods collected, and objective function (2) minimizes the total travel time. Constraint (3) is a degree constraint ensuring that each demand point is visited at most once, constraint (4) preserves the flow conservation, constraint (5) indicates that the vehicles start from the distribution center $v_0$ and finally return to it, constraint (6) is a loading restraint, constraint (7) represents the loading balance after visiting demand points $v_j$, constraint (8) indicates that the vehicle must have a path connection when serving any demand point, and constraints (9) and (10) impose that all variables be 0–1.

4. Model Analysis and Solution

The VRPSDP model, as defined by Equations (1)–(10), is a bi-objective mixed integer optimization programming model. For the bi-objective optimization problem, the $\epsilon$-constraint method proposed by Bérubé can find the complete Pareto frontier [33]. The core idea of this method is to retain only one goal and transform other goals into constraints limited by the value of $\epsilon$. Therefore, according to the basic idea of the $\epsilon$-constraint method, the original problem is transformed into a single-objective problem P0 as follows.

$$f_2=\min {\displaystyle \sum _{i=0}^n{\displaystyle \sum _{j=0}^n{\displaystyle \sum _{k=1}^l{t}_{ij}}}}{x}_{ijk}$$

(11)

subject to constraints (3)–(10):

$${\displaystyle \sum _{i=1}^n{\displaystyle \sum _{k=1}^l{p}_i}}{y}_{ik}\ge \epsilon$$

(12)

This paper designed an approximate solution algorithm based on the $\epsilon$-constraint method. The approximate algorithm first needed to find the TSP (traveling salesman problem) Hamiltonian cycle of the original problem and then arranged the first vehicle to distribute along the TSP Hamiltonian cycle. The vehicle visited one or several demand points in turn. If the vehicle could not meet the following conditions, it returned to the distribution center $v_0$ and let the next vehicle continue to deliver from $v_0$. Condition 1 is that the remaining loading capacity of the vehicle is sufficient to meet the demand of the next customer for picking up and delivering goods; and condition 2 is that if the vehicle visits the next demand point, the actual load after distribution does not exceed its maximum load.

According to the above algorithm idea, the specific steps of the approximate algorithm are given as follows, where $U_k$ represents the loading capacity when vehicle $k$ departs from depot $v_0$.

Time Complexity Analysis of Algorithm 1

The time complexity of using the Christofides algorithm in step 1 is $o(n^3)$. Steps 2 and 3 need to be calculated and compared at most $n$ times. The maximum number of cycles in steps 3–4 is $n$, so the time complexity is $o(n^2)$. The maximum number of cycles in steps 2–4 is $Q$, so the time complexity is $o(Q{n}^3)$. In summary, the time complexity of Algorithm 1 is $o(n^3)$.

The algorithm has been designed, and the time complexity has been analyzed. Next, we analyze the approximation ratio of Algorithm 1. The approximate ratio refers to the ratio between the objective function of the approximate solution and the exact solution. The closer its value is to 1, the closer the solution obtained by the algorithm is to the optimal solution. For example, for a cost minimization problem, let $c(A)$ represent the cost corresponding to Algorithm A and $c(OPT)$ represent the cost corresponding to the optimal solution. If $c(A)\le \alpha c(OPT)$ is true, then $\alpha$ is called the approximate ratio of Algorithm A.

Algorithm 1: The approximate algorithm

STEP 1 Calculate the TSP Hamiltonian cycle through the Christofides Algorithm [34] and record the demand points on the TSP Hamiltonian cycle $\mathrm{as} v_{1^{*}},v_{2^{*}},\dots ,v_{n^{*}}$.

STEP 2$\mathrm{Let} k=1$$. \mathrm{The} \mathrm{shortest} \mathrm{path} S{P}_{{01}^{*}}$ $\mathrm{of} v_0$ $\mathrm{to} v_{1^{*}}$$\mathrm{is} \mathrm{edge} e(v_0,v_{1^{*}})$$. \mathrm{Let} P_k=e(v_0,v_{1^{*}})$$, U_k=d_{1^{*}}$$, \mathrm{and} i=1$.

STEP 3$\mathrm{Let} i=i+1$$. \mathrm{If} i>n$$, \mathrm{proceed} \mathrm{to} \mathrm{STEP}4. \mathrm{Otherwise}, \mathrm{if}Q-{\displaystyle \sum _{j=1}^{i-1}{d}_{j^{*}}}+{\displaystyle \sum _{j=1}^{i-1}{p}_{j^{*}}}\ge p_{i^{*}}-d_{i^{*}}$ $\mathrm{and} U_k+d_{i^{*}}\le Q$$, \mathrm{let} U_k=U_k+d_{i^{*}}$$, \mathrm{add} \mathrm{edge} e(v_{i-1^{*}},v_{i^{*}})$$\mathrm{to} \mathrm{path} P_k$$, \mathrm{and} \mathrm{repeat} \mathrm{STEP} 3. \mathrm{Otherwise}, \mathrm{add} \mathrm{edge} e(v_{i-1^{*}},v_0)$$\mathrm{to} \mathrm{path}{P}_k$$; \mathrm{let} k=k+1$, $U_k=q_{i^{*}}$, $\mathrm{and} P_k=S{P}_{0i^{*}}$; and return to STEP 3.

STEP 4$\mathrm{Add} \mathrm{edge} e(v_{i-1^{*}},v_0)$$\mathrm{to} \mathrm{path} P_k$$, \mathrm{record} \mathrm{the} \mathrm{total} \mathrm{driving} \mathrm{time} T={\displaystyle \sum _{k=1}^{l}t(P_k})$$\mathrm{and} \mathrm{the} \mathrm{total} \mathrm{number} \mathrm{of} \mathrm{goods} \mathrm{retrieved} D={\displaystyle \sum _{k=1}^{l}p(P_i})$, and output the corresponding distribution path of each vehicle.

First, the transportation time corresponding to the optimal solution of the vehicle routing problem is analyzed. The following lemma is derived from the literature [35].

Lemma 1.

The lower bound of the transportation time corresponding to the optimal solution of the vehicle routing problem is$\max \{{\displaystyle \sum _{i=1}^n\frac{2d_{0i}{q}_i}{Q}},T^{*}(TSP)\}$, where$T^{*}(TSP)$is the travel time of the optimal TSP Hamiltonian cycle.

Lemma 1 gives the lower bound of the travel time corresponding to the optimal solution of the vehicle routing problem. Because Problem P0 has additional constraints compared with the VRP problem, its travel time must not be less than that of the general VRP problem. Therefore, the lower bound given by Lemma 1 is also the lower bound of the travel time of Problem P0. Combined with Lemma 1, the following theorem is given.

Theorem 1.

The A* approximation ratio of the algorithm for solving Problem P0 is$\frac{{\displaystyle \sum _{i=1}^{n}Q{d}_{0i}}}{{\displaystyle \sum _{i=1}^n{d}_{0i}{q}_i}}+\beta$, where$d_{0i}$represents the travel time from depot$v_0$to demand point$v_i$, and$\beta$is the approximation ratio of the approximate algorithm for solving the TSP path.

Proof of Theorem 1.

The solution obtained by Algorithm 1 consists of part of the TSP Hamiltonian cycle and several paths connecting the distribution center $v_0$ and the demand point. Here, the part related to the TSP path is called Part 1, and the path connecting the distribution center $v_0$ and the demand point is called Part 2. For Part 1, the overall travel time will not exceed the travel time of the entire TSP path. That is, the maximum transportation cost is $T(STP)$. The maximum value of Part 2 is: ${\displaystyle \sum _{i=1}^n{d}_{0i}}+d_{i0}={\displaystyle \sum _{i=1}^{n}2d_{0i}}$. Then, $\alpha =\frac{c(A^{*})}{c(OPT)}=\frac{{\displaystyle \sum _{i-1}^{n}2d_{0i}}+T(TSP)}{\max \{{\displaystyle \sum _{i=1}^n\frac{2d_{i0}{q}_i}{Q},T^{*}(TSP)\}}}=\frac{{\displaystyle \sum _{i-1}^{n}2d_{0i}}+T(TSP)}{\max \{{\displaystyle \sum _{i=1}^n\frac{2d_{0i}{q}_i}{Q},T^{*}(TSP)\}}}$. If $\beta$ represents the approximation ratio of the approximate algorithm for solving the TSP path, then $\alpha =\frac{{\displaystyle \sum _{i=1}^{n}2d_{0i}+\beta T^{*}(TSP)}}{\max \left\{{\displaystyle \sum _{i=1}^n\frac{2d_{0i}{q}_i}{Q},T^{*}(TSP)}\right\}}$. Next, compare the size of ${\displaystyle \sum _{i=1}^n\frac{2d_{0i}{q}_i}{Q}}$ and $T^{*}(TSP)$. The following two situations exist. If ${\displaystyle \sum _{i=1}^n\frac{2d_{0i}{q}_i}{Q}}\ge T^{*}(TSP)$, then $\alpha =\frac{{\displaystyle \sum _{i=1}^{n}2d_{0i}+\beta T^{*}(TSP)}}{{\displaystyle \sum _{i=1}^n\frac{2d_{0i}{q}_i}{Q}}}=\frac{{\displaystyle \sum _{i=1}^{n}Q{d}_{0i}}}{{\displaystyle \sum _{i=1}^n{d}_{0i}{q}_i}}+\frac{\beta T^{*}(TSP)}{{\displaystyle \sum _{i=1}^n\frac{2d_{0i}{q}_i}{Q}}}\le \frac{{\displaystyle \sum _{i=1}^{n}Q{d}_{0i}}}{{\displaystyle \sum _{i=1}^n{d}_{0i}{q}_i}}+\frac{\beta T^{*}(TSP)}{T^{*}(TSP)}=\frac{{\displaystyle \sum _{i=1}^{n}Q{d}_{0i}}}{{\displaystyle \sum _{i=1}^n{d}_{0i}{q}_i}}+\beta$. If ${\displaystyle \sum _{i=1}^n\frac{2d_{0i}{q}_i}{Q}}\le T^{*}(TSP)$, then $\alpha =\frac{{\displaystyle \sum _{i=1}^{n}2d_{0i}}+\beta T^{*}(TSP)}{T^{*}(TSP)}=\frac{{\displaystyle \sum _{i=1}^{n}2d_{0i}}}{T^{*}(TSP)}+\frac{\beta T^{*}(TSP)}{T^{*}(TSP)}\le \frac{{\displaystyle \sum _{i=1}^{n}2d_{0i}}}{{\displaystyle \sum _{i=1}^n\frac{2d_{0i}{q}_i}{Q}}}+\beta =\frac{{\displaystyle \sum _{i=1}^{n}Q{d}_{0i}}}{{\displaystyle \sum _{i=1}^n{d}_{0i}{q}_i}}+\beta$.

Combining the above two cases, Theorem 1 is proved. □

5. Example

In this section, the performance of the solution method is evaluated. The proposed method in this paper was coded with Java. All experiments were performed on a PC with a 3.3-GHz processor and 8.0-GB RAM under Windows 7.

5.1. Testing Example

In this part, Solomon VRP test set R101 was adopted and modified to generate VRPSDP examples. The first 10 demand points in the database, where node 1, the distribution center, was selected; and the node data are shown in the following table. Without any loss of generality, the vehicle capacity can be assumed to be 100, and the speed of the vehicle can be assumed to be 1 (

Table 1. Testing data (from modified Solomon VRP test set R101).

Node Number	1	2	3	4	5	6	7	8	9	10	11
X Axis	35	41	35	55	55	15	25	20	10	55	30
Y Axis	35	49	17	45	20	30	30	50	43	60	60
Number of Pickup Goods	0	32	17	13	19	25	50	35	10	14	26
Number of Delivery Goods	0	41	12	31	6	7	36	29	40	35	23

).

According to the algorithm designed above, we first find that the TSP path is $v_1\to v_7\to v_6\to v_9\to v_8\to v_{11}\to v_2\to v_{10}\to v_4\to v_5\to v_3$. Second, according to steps 2–4 of the algorithm, the distribution path of each vehicle, the quantity of goods to be collected and the travel time are obtained, as shown in

Table 2. Results of path optimization when the vehicle capacity is 100.

Vehicle	Route	The Number of Collected Goods	The Travel Time
Vehicle 1	$v_1\to v_7\to v_6\to v_1$	75	42
Vehicle 2	$v_1\to v_9\to v_8\to v_{11}\to v_1$	71	78
Vehicle 3	$v_1\to v_2\to v_{10}\to v_1$	46	65
Vehicle 4	$v_1\to v_4\to v_5\to v_3\to v_1$	49	86
Total		241	271

below.

5.2. Sensitivity Analysis

In reality, logistics and distribution companies often have a variety of vehicles, that is, vehicles with different loading capacities, from which to choose. Here, we tested whether a change in the vehicle capacity will have an impact on the results. Continuing to use the above modified Solomon VRP test set R101 example, the vehicle capacity is set to 110, 120, and 150, respectively; and the solution results are shown in

Table 3. Results of path optimization when the vehicle capacity is 110.

Vehicle	Route	The Number of Collected Goods	The Travel Time
Vehicle 1	$v_1\to v_7\to v_6\to v_1$	75	42
Vehicle 2	$v_1\to v_9\to v_8\to v_{11}\to v_1$	71	78
Vehicle 3	$v_1\to v_2\to v_{10}\to v_4\to v_1$	59	70
Vehicle 4	$v_1\to v_5\to v_3\to v_1$	36	63
Total		241	253

,

Table 4. Results of path optimization when the vehicle capacity is 120.

Vehicle	Route	The Number of Collected Goods	The Travel Time
Vehicle 1	$v_1\to v_7\to v_6\to v_9\to v_1$	85	61
Vehicle 2	$v_1\to v_8\to v_{11}\to v_2\to v_1$	93	66
Vehicle 3	$v_1\to v_{10}\to v_4\to v_5\to v_3\to v_1$	63	110
Total		241	237

,

Table 5. Results of path optimization when the vehicle capacity is 150.

Vehicle	Route	The Number of Collected Goods	The Travel Time
Vehicle 1	$v_1\to v_7\to v_6\to v_9\to v_8\to v_1$	120	69
Vehicle 2	$v_1\to v_{11}\to v_2\to v_{10}\to v_4\to v_5\to v_1$	104	124
Vehicle 3	$v_1\to v_3\to v_1$	17	36
Total		241	229

respectively.

The comparison shows that as the vehicle capacity increases, the overall travel time shows a downward trend. The reason is that as the vehicle capacity increases, the constraints loosen, and more distribution routes become feasible paths from which to choose, so the overall travel time is reduced. Generally, it is better to choose a vehicle with a larger capacity than one with a smaller capacity for distribution. However, vehicles with large capacities often correspond to higher transport costs per unit, so they cannot be generalized in reality. Logistics and distribution companies need to weigh and select appropriate vehicles according to the actual situation.

5.3. Extended Testing Example

Table 1 shows an example with 10 demand points. In this part, the range of the demand points was extended to 20. The data of nodes 12–21 are shown in the following 6 (

Table 6. Extended testing data (from modified Solomon VRP test set R101).

Node Number	12	13	14	15	16	17	18	19	20	21
X Axis	20	50	30	15	30	10	5	20	15	45
Y Axis	65	35	25	10	5	20	30	40	60	65
Number of Pickup Goods	23	8	10	21	28	21	17	24	30	48
Number of Delivery Goods	27	11	24	14	35	50	23	18	34	32

). The data of nodes 1–11 and other assumptions are the same as the 5.1 testing example.

According to the algorithm designed above, the distribution path of each vehicle, the quantity of goods to be collected and the travel time are obtained, as shown in

Table 7. Results of path optimization when the vehicle capacity is 100.

Vehicle	Route	The Number of Collected Goods	The Travel Time
Vehicle 1	$v_1\to v_{14}\to v_7\to v_1$	60	37
Vehicle 2	$v_1\to v_6\to v_{18}\to v_{17}\to v_1$	63	71
Vehicle 3	$v_1\to v_{15}\to v_{16}\to v_3\to v_5\to v_{13}\to v_1$	93	112
Vehicle 4	$v_1\to v_4\to v_{10}\to v_{21}\to v_1$	75	80
Vehicle 5	$v_1\to v_2\to v_{11}\to v_{12}\to v_1$	81	76
Vehicle 6	$v_1\to v_{20}\to v_8\to v_1$	65	64
Vehicle 7	$v_1\to v_9\to v_{19}\to v_1$	34	53
Total		471	493

below.

It is shown that the algorithm is also applicable to the solution of more demand points.

6. Conclusions

In this paper, a bi-objective vehicle routing model with simultaneous pickup and delivery minimizing the total travel time and maximizing the total number of collected goods simultaneously was developed. An approximate algorithm based on the $\epsilon$-constraint method was designed. The time complexity analysis showed that the algorithm was a polynomial time algorithm, and the approximation ratio of the algorithm was analyzed.

The model and algorithm were tested by two examples, and the influence of vehicles with different capacities on the travel time is tested by sensitivity analysis. The results showed that as the vehicle capacity increases, the overall travel time showed a downward trend, and a vehicle with a larger loading capacity was better than a vehicle with a smaller loading capacity.

In the future, the research of this paper can still be strengthened through the following work. First, the pickup quantity in this paper is known. In practice, sometimes a vehicle only knows the quantity of delivery goods but does not know the exact pickup quantity before departure; therefore, the pickup quantity can be set to be uncertain in the future to be closer to the actual situation. Second, we can further design an algorithm with a faster solution speed or an approximate algorithm that has a better approximation ratio.