Multi-start iterated local search for the periodic vehicle routing problem with time windows and time spread constraints on services

Abstract

In the field of high-value shipment transportation, companies are faced to the malevolence problem. The risk of ambush increases with the predictability of vehicle routes. This paper addresses a very hard periodic vehicle routing problem with time windows, submitted by a software company specialized in transportation problems with security constraints. The hours of visits to each customer over the planning horizon must be spread in the customer's time window. As the aim is to solve real instances, the running time must be reasonable. A mixed integer linear model and a multi-start iterated local search are proposed. Results are reported on instances derived from classical benchmarks for the vehicle routing problem with time windows, and on two practical instances. Experiments are also conducted on a particular case with a single period, the vehicle routing problem with soft time windows: the new metaheuristic competes with two published algorithms and improves six best known solutions.

Introduction

Despite the dematerialization of financial transactions, made possible by credit cards and more recently by on-line payment systems, cash money is still widely used. Withdrawing money at any time is possible thanks to the ATM (Automated Teller Machines) which are now widespread. The banks subcontract the loading and unloading of these ATM to companies specialized in the transportation of valuable goods. Their mission is to transport valuable items such as coins, bullions, banknotes or jewels from one location to another while protecting them against attacks. The customers of these CIT (Cash In Transit) companies include banks but also supermarkets and other stores. It is therefore a transport activity, but applied to valuable goods, and CIT firms have to plan their daily trips.

Since Dantzig and Ramser [1] introduced the now famous Vehicle Routing Problem (VRP), a large number of variants have been studied. The original purpose is to find a least-cost set of routes to serve from a central depot a set of customers with known demands. In order to better fit real life applications, the basic VRP has been enriched with additional constraints, e.g., time windows to visit customers, pick-up and delivery operations, heterogeneous fleet of vehicles, etc. Two good reviews on vehicle routing problems and their various applications can be found in the books of Toth and Vigo [2] and Golden et al. [3]. As the tasks of CIT companies are typically planned over several days (e.g., one week) and most customers must be visited daily within specified time windows, the transport operations can be modeled here as a periodic vehicle routing problem with time windows.

However, the activity considered is inherently dangerous, with potentially catastrophic outcomes, thus careful planning and preparation are essential. In addition to traditional transportation constraints, the nature of the goods raises security issues rarely studied in vehicle routing. During route construction, it is important to take into account the risk dimension. The risk is often defined as the combination of the probability that an event occurs with the expected loss. The damage in case of attack can be limited by armored vehicles, weapons and adequate training of conveyors. As the attack probability increases with the foreseeable nature of the routes, a complementary measure is to make the trips less predictable, which leads to the following question: How to design vehicle routes that look “random”?

A possible option is to consider that the sequencing of customers must vary over successive periods, leading to edges-disjoint routes. This strategy favors the protection of vehicles and conveyors while they are on the road but, according to the experience of security companies, the vulnerability is maximal when the vehicle is parked to serve a customer. As it is impossible to avoid these stops, another option is to forbid arrivals before the beginning of the customer's time window and to make the arrival times irregular over successive periods.

The problem addressed in this paper is a periodic vehicle routing problem with no-wait time windows and the second option, i.e., irregular arrival times. As minimizing the cost of routes and dispersing arrival times are two conflicting objectives, the total cost of the routes is minimized, subject to the constraint that the hours of any two visits to the same customer must differ by a given time constant $ϵ$. We call this problem the periodic vehicle routing problems with time spread constraints on services (PVRPTS). The PVRPTS is extremely hard. For instance, the relocation of a customer in a different route, even in the same period, can affect arrival times in all other periods. Moreover, deciding if a move is feasible or not and evaluating its cost variation are complicated by the time windows and the prohibition of waiting times.

We found no published reference on the PVRPTS but provide a review on vehicle routing problems related to security constraints in Section 2. The problem is formally defined and modeled as a mixed integer linear problem in Section 3. A metaheuristic with multiple restarts, based on iterated local search, is described in Section 4. The linear model and the metaheuristic are tested in Section 5 on classical Vehicle Routing Problem with Time Windows (VRPTW) instances (modified with the time spread constraint) and on two real instances provided by a French security company. The proposed method is also tested on a single-period particular case, the VRP with soft time windows (VRPSTW), and compared with state-of-the-art heuristic from the literature. Section 6 is finally devoted to concluding remarks and perspectives.