Variable neighborhood search for consistent vehicle routing problem

Highlights

• A new algorithm for ConVRP is proposed. It outperforms extant algorithms for ConVRP.

• Several techniques are used to reduce computational time of iterated local search.

• An effective shaking method is proposed.

• Time differences of customer points are calculated using a new method.

Abstract

This article presents a variable neighborhood search (VNS) algorithm for the consistent vehicle routing problem (ConVRP). ConVRP is a variant of the vehicle routing problem (VRP). In ConVRP, vehicle routes must be designed for multiple days, and each customer must be visited by the same driver at approximately an identical time on each day. VNS is an efficient algorithmic framework and is widely used. The proposed algorithm consists of two stages. In the first stage, VNS is applied to obtain approximately optimized solutions. The solutions obtained might be infeasible. If a solution is of acceptable quality, the second stage is applied to make it feasible and optimize it further. Several techniques are employed to reduce computation time of the local search stage. A special shaking method is introduced and proofed to be more effective than ordinary methods by experiments. A new method for computing time difference excess is proposed to solve the problem that change of time difference excess caused by operations on an individual day is not obvious. The proposed algorithm is tested on the benchmark ConVRP data set and compared with extant ConVRP approaches from the literature. The results demonstrate that VNS outperforms all the extant ConVRP approaches in terms of quality of solutions obtained.

Introduction

The consistent vehicle routing problem (ConVRP) was proposed by Groër, Golden, and Wasil (2009). It is used to model delivery activities of small packages. In small package shipping industry, service consistency is important for improving customer satisfaction. ConVRP is an extension of the periodic vehicle routing problem (PVRP)(Campbell & Wilson, 2014). PVRP involves managing delivery activities over a period. In typical cases, a period consists of several days, and the routes for delivery must be designed for each day. On a particular day, a customer may or may not require service; therefore, the routes designed for different days generally vary. As there is negligible interplay between any two days, PVRP can be split into several separated capacitated vehicle routing problems (CVRPs)(Xiao, Zhao, Ikou, & Mladenovic, 2014). ConVRP takes customer satisfaction into consideration. From the perspective of a customer, if a single driver provides the service, the driver must be familiar with the situations in the vicinity, e.g., the traffic situation around the district, and thus provide efficient service. In addition, if the service is provided at an approximately similar time on each day, it will be more convenient for the customer to arrange his/her schedule for the service. Based on these factors, ConVRP adds two constraints to PVRP: first, each customer must be served by the same driver over the planning horizon; second, the difference between the earliest time and the latest time that a customer receives service on all days must not exceed a specified limit. The first constraint is called service consistency, and the second is called time consistency.

The mixed integer program of ConVRP was introduced by Groër et al. (2009). For the integrity of description, we present the program in this section. ConVRP is defined on a complete graph $G=(N,A),$ in which $N=\{0,1\cdots n\}$ is the set of customers, {0} is the departure point of vehicles, and $A=\left\{\right(i,j\left)\right|i,j\in N,i\ne j\}$ is the set of arcs. Each arc (i, j) is associated with a travel time t_ij. $K=\{1,2\cdots m\}$ is the set of vehicles; the number of vehicles is not limited. The planning horizon consists of |D| days, in which $D=\{1,2\cdots |D\left|\right\}$ is the set of days. On each day d, each customer i is associated with a demand q_id and service time s_id. $q_{id}=0$ indicates that customer i has no demand and will not be served on day d. On each day, all the vehicles depart from point 0 at time 0 and return to point 0 after visiting customers they serve. Each customer that has demand on this day must be served exactly once. The demand of all the customers must be fulfilled. All the vehicles have identical capacity Q and can operate for no more than T units of time on each day. The time that a customer receives service is the time at which the vehicle arrives at this customer’s location. a_id is the arrival time at customer i on day d, it equals 0 if customer i is not served on day d ($i=0$ is the depot). $w_{id}=1$ if customer i requires service on day d and $w_{id}=0$ otherwise (the depot requires service on all days). L is the time difference limit, which limits the difference between the earliest time and the latest time that a customer receives service. Decision variable x_ijkd equals 1 if vehicle k visits customer j immediately after customer i on day d and equals 0 otherwise. Decision variable y_ikd equals 1 if customer i is visited by vehicle k on day d and equals 0 otherwise. The objective function and constraints are as follows:

$$\begin{array}{c}\hfill \min \sum _{d\in D}\sum _{k\in K}\sum _{i\in N}\sum _{j\in N}{t}_{ij}{x}_{ijkd}\end{array}$$

Subject to:

$$y_{0kd}=1\forall k\in K,d\in D$$

$$a_{0d}=0\forall d\in D$$

$$\sum _{k\in K}{y}_{ikd}=w_{id}\forall i\in N\setminus \left\{0\right\},d\in D$$

$$\begin{array}{ccc}\hfill \sum _{i\in N\setminus \left\{0\right\}}{q}_{id}{y}_{ikd}\le Q& & \hfill \forall k\in K,d\in D\end{array}$$

$$\begin{array}{ccc}\hfill \sum _{i\in N}{x}_{ijkd}=\sum _{i\in N}{x}_{jikd}=y_{jkd}& & \hfill \forall i,j\in N,k\in K,d\in D\end{array}$$

$$\begin{array}{ccc}& & w_{ia}+w_{ib}-2\le y_{ika}-y_{ikb}\le -(w_{ia}+w_{ib}-2)\hfill \\ & & \forall i\in N,k\in K,a,b\in D,a\ne b\hfill \end{array}$$

$$\begin{array}{ccc}& & a_{id}+x_{ijkd}(s_{id}+t_{ij})-(1-x_{ijkd})T\le a_{id}\hfill \\ & & \forall i\in N,j\in N\setminus \left\{0\right\},k\in K,d\in D\hfill \end{array}$$

$$\begin{array}{ccc}& & a_{id}+x_{ijkd}(s_{id}+t_{ij})+T(1-x_{ijkd})\ge a_{id}\hfill \\ & & \forall i\in N,j\in N\setminus \left\{0\right\},k\in K,d\in D\hfill \end{array}$$

$$\begin{array}{ccc}\hfill \sum _{i,j\in S}{x}_{ijkd}\le \left|S\right|-1& & \hfill \forall S\subseteq N,2\le \left|S\right|\le n,k\in K,d\in D\end{array}$$

$$\begin{array}{ccc}\hfill 0\le a_{id}+w_{id}(s_{id}+t_{i0})\le T{w}_{id}& & \hfill \forall i\in N\setminus \left\{0\right\},d\in D\end{array}$$

$$\begin{array}{ccc}& & -L+T(w_{ia}+w_{ib}-2)\le a_{ia}-a_{ib}\le L-T(w_{ia}+w_{ib}-2)\hfill \\ & & \forall i\in N,a,b\in D,a\ne b\hfill \end{array}$$

$$\begin{array}{ccc}\hfill x_{ijkd}\in \{0,1\};y_{ijkd}\in \{0,1\};a_{id}\ge 0& & \hfill \forall i,j\in N,k\in K,d\in D\end{array}$$

The objective function (1) minimizes the total travel time of all vehicles over all days. By constraints (2) and (3), the depot is visited at time 0 by all vehicles on all days. Constraint (4) ensures that customers are visited exactly once when they require service. Constraint (5) guarantees that each vehicle carries no more than Q units on any day. By constraint (6), each customer has only one predecessor and one successor. Constraint (7) ensures that each customer is served by one driver. Constraints (8) and (9) determine the arrival times at customers. Constraint (10) eliminates subtours. Travel time limit is defined in constraint (11). Constraint (12) ensures that the difference between arrival times at each customer on any two days is no more than L, if this customer requires service on both two days.

The structure of this paper is as follows. In Section 2, we give a relevant literature review. In Section 3, we describe our VNS algorithm. In Section 4, we present experiments that proof the efficiency of VNS and the other conclusions. In Section 5, the conclusions are summarized.