A template-based Tabu Search algorithm for the Consistent Vehicle Routing Problem

Abstract

This paper presents a generic template-based solution framework and its application to the so-called Consistent Vehicle Routing Problem (ConVRP). The ConVRP is an NP-hard combinatorial optimization problem and involves the design of a set of minimum cost vehicle routes to service a set of customers with known demands over multiple days. Customers may receive service either once or with a predefined frequency; however frequent customers must receive consistent service, i.e., must be visited by the same driver over approximately the same time throughout the planning period. The proposed solution framework adopts a two-level master–slave decomposition scheme. Initially, a master template route schedule is constructed in an effort to determine the service sequence and assignment of frequent customers to vehicles. On return, the master template is used as the basis to design the actual vehicle routes and service schedules for both frequent and non-frequent customers over multiple days. To this end, a Tabu Search improvement method is employed that operates on a dual mode basis and modifies both the template routes and the actual daily schedules in a sequential fashion. Computational experiments on benchmark data sets illustrate the competitiveness of the proposed approach compared to existing results.

Introduction

The design and implementation of periodic delivery systems has become a crucial concern for modern companies seeking to provide high quality and low cost services to their customers. Evidently, optimizing repetitive delivery operations over multiple days can add up to significant cost savings, and thus improve productivity and competitiveness. Periodic deliveries occur in a wide range of real life applications, including among others refuse and municipal waste collection, mail collection and delivery, scheduled retail and wholesale delivery and distribution, vending machine replenishment, and elevator repair and maintenance (Francis, Smilowitz, & Tzur,Francis et al.2 22000088). Therefore, in practical terms studying such operational planning problems definitely seems worthwhile, apart from the theoretical and computational research challenges arising due to their combinatorial nature.

In broad terms, multi-period vehicle routing and scheduling problems deal with the optimum assignment and service sequence of a set of customer orders to a fleet of vehicles over multiple days (the term “day” is used as a general unit of time throughout this paper). In the literature, these problems are typically modeled as Periodic Vehicle Routing Problems (PVRP) (Beltrami & Bodin, 1974). The PVRP is a generalization of the well-known Capacitated Vehicle Routing Problem (CVRP) (Tarantilis, 2005) and involves the design of a set of vehicle routes, over a predefined planning horizon, in order to service a set of customers with known demand and frequencies of service (i.e. customers can be visited according to different day combinations). Typically, the objective is to minimize the total traveling cost, expressed in terms of one-time (e.g. fleet size) and recurring costs (e.g. distance traveled), while satisfying operational constraints (e.g. vehicle capacity and visit requirements). Note that besides the daily routing decisions (i.e. assignment and service sequence of each customer on vehicle routes), a schedule from a candidate set of schedules for each customer must be also selected.

Contrary to the above described periodic delivery operational setting, in recent years more and more companies tend to focus and invest on brand loyalty and customer relationship management, and thereafter, they are interested in implementing customer-oriented rather than demand-oriented approaches. For example, there are numerous real life applications in which customers need to be visited by the same service provider (i.e. vehicle crew and driver). Furthermore, in many cases customers need to be serviced according to a predefined visiting sequence or a certain service time consistency (e.g. a minimum variance of service times over multiple days). Typical real life paradigms that depict these type of consistent service considerations can be found in parcel deliveries and courier services, home care and nursing services for the elderly and cleaning services. In these cases, the main effort is to gain competitive advantage by forming bonds with the customers.

A first attempt in the literature towards modeling and solving multi-period vehicle routing and scheduling problems with consistent customer service constraints has been put forward by Groër, Golden, and Wasil,Gro;r et al. (2009). In specific, they introduced the so-called Consistent Vehicle Routing Problem (ConVRP), motivated by a real-life parcel delivery application. To this end, a multi-start solution construction framework is proposed, combined with a Record-to-Record travel local search metaheuristic algorithm (Li, Golden, & Wasil,Li et al.2 22000055) and savings-based construction heuristics.

The ConVRP is an NP-hard combinatorial optimization problem in the strong sense and involves the design of minimum cost routes in order to service a set of customers with known demand over multiple days via a homogeneous fleet of depot returning capacitated vehicles. Customers may receive service either once (non frequent customers) or with a predefined frequency (frequent customers); however frequent customers must receive consistent service throughout the planning period, such that (a) their visiting sequence remains the same (or similarly the maximum service time difference between the earliest and latest vehicle arrival times over multiple days does not exceed a maximum time limit) and (b) the service is performed by the same vehicle. On the other hand, during each day each customer must be visited only once by exactly one vehicle, while each vehicle has a maximum carrying capacity and operates for no more than a maximum time limit. The goal is to minimize the total distance traveled by the vehicles such that all customer requirements are fulfilled without violating capacity, route duration and consistent service constraints.

Towards this new and emerging line of research, the main contribution and aim of this paper is to design and develop a generic and flexible template-based solution framework for the ConVRP. Following the concept of template route schedules, the proposed approach adopts a two-level decomposition scheme and solves a master and a slave sub-problem in a sequential fashion. In particular, the master sub-problem is concerned with the design of a template route schedule in order to determine the assignment of frequent customers to vehicles and their visiting sequences, while the corresponding slave sub-problem seeks to determine the actual daily vehicle routes for both frequent and non-frequent customers on the basis of the template route schedule. From the implementation viewpoint, effective savings and insertion based construction heuristics are utilized, coupled with a Tabu Search (TS) algorithm that operates on a dual mode basis. Given an initial template route schedule, TS is applied in an effort to minimize the total distance traveled, considering only frequent customers (master mode). On return, the corresponding daily schedules for both frequent and non-frequent customers are constructed and further improved by TS (slave mode). In this case, TS is applied to the actual daily schedules and takes into account, apart from vehicle capacities and route duration restrictions, the precedence constraints (i.e. assignment and visiting sequence requirements) dictated by the corresponding template route schedule.

For the evaluation of the proposed template-based solution approach, computational experiments on benchmark data sets of the literature are reported. Compared to existing results, it proved to be highly competitive and improved the best reported cumulative and mean results over all problem instances with very reasonable computational requirements for practical applications. To this end, competitive advantage of proposed solution framework is its fairly simple algorithmic structure, its flexibility to accommodate various types of consistent service constraints and the small number of parameters introduced.

The remainder of this paper is organized as follows. Section 3 presents the proposed solution framework and provides detailed descriptions of all algorithmic components and mechanisms. Computational experiments assessing the quality of the proposed approach along with a comparative performance analysis are presented in Section 4. Finally, in Section 5 conclusions are drawn and pointers for future research are provided.