^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

A multidepot VRP is solved in the context of total urban traffic equilibrium. Under the total traffic equilibrium, the multidepot VRP is changed to GDAP (the problem of Grouping Customers + Estimating OD Traffic + Assigning traffic) and bilevel programming is used to model the problem, where the upper model determines the customers that each truck visits and adds the trucks’ trips to the initial OD (Origin/Destination) trips, and the lower model assigns the OD trips to road network. Feedback between upper model and lower model is iterated through OD trips; thus total traffic equilibrium can be simulated.

The VRP is a generic name referring to a class of combinatorial optimization problems in which a number of vehicles serve the customers. The vehicles leave the depot, serve customers in the network, and return to the depot after completion of their routes. Dantzig and Ramser (1959) first proposed this problem in the literature. VRP is generally defined by a graph

Travel cost (

Currently, based on partial traffic equilibrium, some researches calculate the shortest path travel time in delivery time windows dynamically. For example, in recent years, several work [

When location of depots/facilities and demand flows between depots and facilities are stable, VRP should be solved statically (a priori) rather dynamically. In this case, the routing behavior of delivery trucks is similar to other vehicles. Both take traffic condition into account. For example, delivery trucks will choose the roadways with less traffic, and other vehicles may keep away from the roadways where the traffic is heavily affected by the travelling and unloading of delivery trucks. It means that interactions exist among all vehicles and traffic flows on roadways result from all drivers’ choice of their path. Then, travel times on roadways are determined by links’ capacities and the corresponding traffic flows. Moreover, because different VRP schemes will result in different OD traffic of delivery trucks, the delivery loops/paths also interact with urban OD traffic. Therefore, it can be said that delivery loops/paths, which are obtained under the assumption that other vehicles are not affected by delivery trucks, are not optimized for the real situation.

When the influence of delivery trucks on road traffic cannot be ignored, the VRP should be treated as a traffic problem at macro level rather than a logistics issue at micro level. Then, we must design the delivery loops/paths from the view of total traffic equilibrium, namely, considering the interactions among all vehicles.

In the real world, the phenomena that the delivery trucks give no influence on road traffic in a city have been disappearing because (1) road capacity in most large cities (especially Chinese ones) is at the critical point where a few additional vehicles may cause severe congestion in some roadways and further result in the change of the whole urban road traffic pattern and (2) rapid developed e-commerce changes citizen’s shopping behavior. Online shopping reduces personal travel but increases delivery truck traffic. Currently, the delivery trucks have become a significant part of overall urban traffic, which is shown in Figures

Urban traffic condition.

Due to the above reasons, the VRP is no more just the problem that assigns customers and travel paths for delivery trucks based on a given OD traffic. One should consider delivery and other vehicles as a whole when studying their path choice behaviors. Thus, one should take the interactions between the delivery trucks and other vehicles into account directly because delivery schemes will change OD trips and then the roadway traffic, while the OD trips and roadway traffic inversely determine the VRP schemes. This is a solution method of VRP in terms of total traffic equilibrium. Here, the “total” traffic equilibrium means the interactions between delivery trucks and other vehicles are considered directly. Under the “total” user equilibrium, neither delivery trucks nor other vehicles have willingness to change their travel paths.

Therefore, in terms of total traffic equilibrium, the VRP becomes GDAP (problem of Grouping Customers + Estimating OD Traffic + Assigning Traffic), among which “Grouping Customers” means the works that customers are assigned to different depots, and then a multidepot VRP is solved by transforming into a group of single-depot VRPS; “Estimating OD Traffic” means the works that the trip chains of the deliveries trucks are added to the given

We think the delivery schemes (namely, the being served customers and travel paths of the trucks) should be obtained by solving the three equations simultaneously. Here, first equation is VRP model, the second equation is traffic assignment model, and the third one is the method to put delivery truck trip chains into the OD matrix.

We may use the bilevel programming to model above problem, where the interactions between delivery trucks and other vehicles are considered. The upper model determines the customers that each delivery truck visits and the path of each delivery truck, then we add these delivery trucks OD trips into the initial OD matrix, and the lower model is a traffic assignment model that finds paths for all the vehicles. Interactions between the LM and UM are iterated through updating OD. Thus, the total traffic equilibrium on a road network can be simulated and loops/paths of delivery trucks can be found in convergent outputs.

The rest of the paper is organized as follows. In Section

Over the last three decades, the number of academic publications on the numerous variants of the VRP has increased extensively [

After Dantzig and Ramser (1959), a considerable number of variants of VRP have been studied, including (1) the VRP with hard, soft, and fuzzy service time windows (VRPTW); (2) the VRP considering backhauls (VRPB); (3) the VRP considering maximum route length, (DVRP); (4) periodic VRP (PVRP); (5) VRP with multiple trips (VPRMT); (6) split delivery vehicle routing problem (SDVRP) and others; (7) the VRP with minimized emissions [

Different variants of the VRP.

Since many variants of VRP are NP-hard problems, lots of researchers have proposed solution algorithms. For example, Clarke and Wright [

At present, solution of VRP is not only vital in the design of distribution systems in supply chain management, but also important in urban solid waste collection, street cleaning, school bus routing, routing of salespeople, and courier services. Researches can be roughly divided into theoretical papers providing mathematical formulations and exact or approximate solution methods for academic problems and case-oriented papers. More recently, attention has been devoted to more complex variants of the VRP (usually called “rich” VRPs) that are closer to the practical distribution problems than classic VRP models.

There is also literature which puts VRP in real-world context to consider the dynamics of travelling times on an urban road network because in urban areas the travel speeds (or times) typically vary during the day due to differing congestion patterns. Malandraki and Daskin [

Taniguchi and Yamada [

Çetinkaya et al. [

Real-world applications of VRP often include two important dimensions: evolution and quality of information [

In real-world applications, static design is more important when VRP is between the layers of depots and facilities. In a small period of time (e.g., a quarter or a month), spatial distribution of depots/facilities, demand flows between depots and facilities of the urban solid waste collection, street cleaning, school bus routing, routing of salespeople, and courier services may hardly change. Thus one delivery scheme may be used for the whole quarter or month. This is similar to bus transit service, where daily personal travel demand is relatively stable and the redesign of bus routes day by day and the dynamic scheduling of buses hour by hour are not necessary.

Taniguchi et al. [

Researchers in Taniguchi team also adopted bilevel programming model to integrate a supply chain network (SCN) with a transportation network (TN) in terms of traffic equilibrium. Among them, Yamada et al. [

When location of depots/facilities and demand flows between depots and facilities are stable, VRP should be solved statically (a priori) rather dynamically. In this case, the routing behavior of delivery trucks is similar to other vehicles. Both take traffic condition into account. For example, delivery trucks will choose the roadways with less traffic, and other vehicles may keep away from the roadways where the traffic is heavily affected by the travelling and unloading of delivery trucks. It means that interactions exist among all vehicles and traffic flows on roadways result from all drivers’ choice of their path. Then, travel times on roadways are determined by links’ capacities and the corresponding traffic flows. Moreover, because different VRP schemes will result in different OD traffic of delivery trucks, the delivery loops/paths also interact with urban OD traffic. Therefore, it can be said that delivery loops/paths, which are obtained under the assumption that other vehicles are not affected by delivery trucks, are not optimized for the real situation.

When the influence of delivery trucks on road traffic cannot be ignored due to the recurring congestion and sophisticated e-commerce business models [

Our contributions in this study are as follows.

(1) Under the total traffic equilibrium, transforming the multidepot VRP to GDAP (the problem of Grouping Customers + Estimating OD Traffic + Assigning traffic) to take the interactions between delivery trucks and other vehicles into account to obtain delivery schemes under “the total traffic equilibrium.”

(2) Solving GDAP with bilevel model. Based on the feedback loop of “the problem of Grouping Customers - determining the delivery routes - updating OD traffic - assigning OD traffic – re-grouping…,” firstly the customers are divided into several groups and secondly the delivery loops/paths for the groups are obtained and the initial OD matrix is updated, and thirdly links’ traffic flows are calculated by user equilibrium traffic assignment model.

(3) Carrying out a case study with actual data in Dalian to examine and verify the method and provide some useful findings.

(4) Evaluating the delivery schemes based on not only the delivery time but also the traffic situation of the entire road network.

(A1) Study area consists of continuous but nonoverlapping traffic zones, the OD trips of vehicles other than delivery trucks do not change, but the paths between origins and destinations are not fixed, which will be determined based on UE Theory.

(A2) Demand of each costumer is given.

(A3) Depots’ supply amounts are big enough.

(A4) Length of delivery loop is shorter than the truck’s maximum range.

(A5) All delivery trucks are the same type, with the loading capacity given.

(A6) Loading and discharging times during the delivery are ignored.

(A7) One truck is equivalent to 3 per car units.

(A8) Drivers know the travel times of all roadways and try to choose the shortest path.

The variables are defined as follows:

The upper model is as follows:

Equation (

Actually,

The variables used are defined as follows:

The lower model is as follows:

Equation (

The model can be solved by iterative calculation of “Grouping Customers → Determining the delivery schemes → Changing OD matrix → Assigning OD traffic → Re-Grouping Customers → ⋯.” Here we use the network in Figure

Example of an area with 3 depots and 5 costumers.

We optimize the delivery scheme in the context of fixed links’ travel speeds for the multidepot VRP by Generic Algorithm (GA). Firstly, the multidepot VRP is transformed into several single-depot VRPs by grouping the customers. Due to the grouping,

The grouping works are as follows.

Calculate

Find the minimum

Compare the minimum

Assign the customers of Group 4 to Groups 1–3 through enumeration method, and obtain

Here,

The delivery schemes are optimized as follows.

Calculating

Code and generate the initial population, namely, vehicle routes.

Examine the feasibilities of each individual according to the constraints.

For the feasible individuals, calculate their fitness.

Perform selection and mutation operations.

Determine terminating the calculation.

Perform crossover and mutation operations, and return to Step 2.

For each step, the detailed calculations are as follows.

(a) Count customers in Group 1 in

(b) Determine the insertion times of ID of Depot 6 to get

(c) Add “6” to the head and end of Array 2 to ensure the truck starting and returning to Depot 6. Get the final array for Group 1 (Array 3).

(d) Repeat the above works for Group 2 and Group 3.

(e) Group the final arrays of Groups 1, 2, and 3 in turn to get an individual set of the initial population. With this method,

For example, the initial population is coded by natural coding method, and each chromosome is encoded by three Gene Segments (Gene Segment 1, Gene Segment 2, and Gene Segment 3), which represent the code of Groups 1, 2, and 3, respectively. In the case of Chromosome [6 2 1 6 7 4 7 8 3 8 5 8], (6 2 1 6) is

Determine the OD trips of the delivery trucks based on the optimal grouping pattern, and then add the OD trips of the delivery trucks to the former OD matrix.

Frank-Wolfe (FW) approach is used to solve the lower model, which is a normal user equilibrium traffic assignment model, as follows.

Set

Set

Repeat all-or-nothing assignment with

Solve

One has

If

In this paper, the main model (the upper one) is proposed to describe VRP. The decision variable for the upper model is

The retailing delivery of aquatic products in the Xigang district in Dalian (China) is used to do the case study. As shown in Figure

The demands of the customers (Ton/Day).

Costumer | 1 Wal-Mart | 2 METRO | 3 Tesco | 4 Shunming | 5 Daqing | 6 Zhongbei | 7 New Mart |
---|---|---|---|---|---|---|---|

(Zhongshan Rd) | (Shugang Rd) | (Changchu Rd) | Supermarket | Supermarket | Supermarket | (Yuouyi Street) | |

Amount | 0.19 | 0.17 | 0.18 | 0.21 | 0.18 | 0.21 | 0.18 |

Costumer | 8 Dalian seafood | 9 Champs | 10 Future | 11 Tesco | 12 Dashang | 13 Wal-Mart | 14 Tesco |

Supermarket | Supermarket | Supermarket | (Changchun Rd) | (Changchun Rd) | (XiAn Rd) | (Victory Rd) | |

Amount | 0.2 | 0.15 | 0.21 | 0.18 | 0.16 | 0.18 | 0.17 |

Costumer | 15 Tesco | 16 Carrefour | 17 Carrefour | 18 Hualian | 19 Wilson | 20 Sanbao | 21 Lihua |

(Jiefang Road) | (Huanghe Rd) | (Changjiang Rd) | Supermarket | Supermarket | Supermarket | Supermarket | |

Amount | 0.15 | 0.15 | 0.18 | 0.2 | 0.19 | 0.2 | 0.21 |

Costumer | 22 Yongfu fresh | 23 Tesco | 24 All Poly | 25 YiFeng | 26 Food | 27 Triumph | |

Supermarkets | (Baishan Rd) | supermarkets | Supermarket | Supermarket | Seafood Mall | ||

Amount | 0.2 | 0.15 | 0.09 | 0.21 | 0.1 | 0.32 |

Traffic zones, depots, costumers, and road network.

The study area is divided into 31 traffic zones (Figure

The solution of the optimization model, which is the delivery schemes under total traffic equilibrium, is shown in Figure

Delivery loops and travel paths.

To demonstrate the validity of the proposed model, we set some scenarios by changing OD matrices and the numbers of delivery trucks, respectively and then do sensitivity analysis. For building the scenarios, we adjust vehicle OD trip matrix of Dalian in 2011 with

The indices are shown in Figure

When

Delivery routes from Depot 1 to Customer 21 by two methods.

When

Delivery paths and service depots of Customer 8 in the two schemes.

When

Based on the above analyses, we can understand that the delivery schemes under total traffic equilibrium change as the changes of the traffic volume. In case of free flow, the trucks travel along the loops/paths with the shortest travel distance (or time). As the traffic volume increment the path with the shortest road distance may not be that with the shortest travel time. According to the congestion degrees, some other paths may become the shortest travel time ones. Therefore, the delivery schemes under total traffic equilibrium are practical.

Furthermore, the traffic of delivery trucks may also change the travel behaviors of other vehicles. To demonstrate this, we set some scenarios by setting the OD matrix of other vehicles as

The obtained results are shown in Figure

Average travel times of other vehicles.

In many Chinese large cities, in order to manage traffic, trucks are banned from travelling in the morning and evening rush hours. With the information in Figure

It is the green point in Figure

It is the blue triangle in Figure

It is the red square in Figure

In addition, VRP schemes based on partial traffic equilibrium may worsen the service level of the whole road network and cause severe congestion, because some roadways on the delivery paths may already have large amount of traffic to be at saturation, and a few delivery trucks may lead to severe congestion. Figure

Road traffic flows with delivery vehicles under different equilibriums.

Traffic flow under partial equilibrium

Traffic flow under total equilibrium

Moreover, the results also show that in the situation of the total equilibrium the travel flow and travel speed on Link 2 are 1699 pcu/h and 13.4 km/h. However, in the situation of partial equilibrium, the corresponding figures are 1746 pcu/h and 12.5 km/h, respectively. It proved that without taking them into account, more delivery trucks will choose Link 2, which will be more congested after the delivery trucks join in. Thus, the shortest path under partial equilibrium on it is a fake one.

In terms of calculation time for the solution, for the partial traffic equilibrium method it is 13 seconds and for the total traffic equilibrium method it is 256 seconds (CUP: Core i5, 1.5 G). It is obvious that to simulate the interactions between all vehicles and between delivery schemes and OD trips, iterative computations should be done; thus our method costs more time for solution calculation. However, since GDAP is not a dynamic problem but prior simulation of total traffic equilibrium with bilevel model and real-time calculation is not needed, the computing speed and efficiency are not important.

In this paper, we change multidepot VRP into GDAP, which is an alternative way of designing routes for a multidepot VRP and then modelling urban goods transport. A bilevel programming is proposed to model the GDAP, and the interaction among the path choice behaviors of all vehicles and the interaction between delivery schemes and OD trip matrix can be simulated. Then the solution for multidepot VRP is obtained by hybrid grouping method, genetic algorithm, and Frank-Wolf algorithm. The delivery schemes outputs from the model are those under total traffic equilibrium and thus are realistic, which can make full use of road capacity and balance the traffic flow in the entire network.

Compared with the solution under partial traffic equilibrium, the solution under the total traffic equilibrium can not only shorten the delivery time, but also help to avoid traffic congestions on the roads which are near saturation. The bilevel model has good adaptability for optimizing MDVRP. It can be used as a reference for further study on combination optimization problems.

We recommend that further research be done comparing our new method with other methods for real life situations. More research could also be done on algorithms for cases where loads on trucks are not constant.

The limits of the method are that (1) it is difficult to get the OD of other vehicles; (2) the GA cannot assure the global optimal solution of the delivery schemes.

The authors declare that they have no conflicts of interest.

This research is supported by the key project of Natural Science Foundation of China (Grant no. 71431001), youth project of Natural Science Foundation of China (Grant no. 71402013), and Humanity and Social Science Youth Foundation of Ministry of Education of China (14YJC630010).