Reactive Variable Neighborhood Tabu Search for Heterogeneous Fleet Vehicle Routing And

Reactive Variable Neighborhood Tabu Search for Heterogeneous Fleet Vehicle Routing and Scheduling Problems

D.C. Paraskevopoulos, P.P. Repoussis, C.D. Tarantilis, G. Ioannou and G.P. Prastacos Department of Management Science and Technology Athens University of Economics and Business

1. Introduction

Effective distribution logistics is the main target of modern companies, thriving for excellent customer service with minimal investment and operational costs. Fuel consumption, equipment acquisition and maintenance, personnel wages, all related to transportation operations constitute a significant part of logistics costs for most firms [1]. Even small efficiency improvements, related to resource utilization in a daily basis, can lead to huge improvements in the long term.

Broadly speaking, vehicle routing and scheduling problems involve the optimum assignment and sequencing of a set of transportation orders to a fleet of vehicles, considering various operational constraints. Such constraints have particular importance in those real-life settings where the vehicle fleet is heterogeneous, i.e., vehicles differ in their equipment, capacity, age or cost [2]. The use of a heterogeneous fleet of vehicles has multiple advantages. The scheduler can revise the fleet composition to better suit customer needs because they may require vehicles with expensive equipment [3]. It is also possible to service customers requiring small vehicles because of accessibility restrictions in urban areas, environmental concerns or physical restrictions on the vehicle size and weight [4]. Furthermore, vehicles of different carrying capacities give the flexibility to allocate capacity according to the customer’s varying demand, in a more cost effective way, by deploying the appropriate vehicle types to areas with the analogous concentration of customers [5, 6].

Formally, the Heterogeneous Fleet Vehicle Routing Problem with Time Windows (HFVRPTW) involves the design of a set of minimum cost routes, each originating and terminating at the depot, using a heterogeneous fleet of vehicles with fixed and variable costs, to service a set of customers with known demands. Each customer must be serviced only once, within a predefined time window, by exactly one vehicle, while the total demand of a route must not exceed the capacity of the vehicle type assigned to it. The distribution cost of a vehicle derives from the sum of its fixed cost and a variable cost proportional to the distance travelled.

Given the above specifications, two major variants are treated in this study. The first one is known as the Fleet Size and Mix VRPTW (FSMVRPTW) ([7, 8]), and the second as the Heterogeneous Fixed Fleet VRPTW (HFFVRPTW). The FSMVRPTW involves various types of vehicles of unlimited availability. The objective is to determine the optimal fleet composition with minimum overall distribution costs (the sum of fixed and variable costs). Going a step further, the HFFVRPTW generalizes FSMVRPTW by limiting the number of available vehicles of each type that can be deployed [2].

Due to their wide applicability and high complexity, vehicle routing problems with heterogeneous fleet have attracted substantial research efforts. In particular, the HFVRPTW is NP–hard in the strong sense (since it generalizes the VRPTW), and the literature includes both exact ([2], [9]) and heuristic solution approaches for these problems. However, large scale problem instances can be intractably hard to solve to optimality. For this reason, the focus of most research is on the design and implementation of metaheuristic approaches that produce high quality solutions within reasonable computational times, including among others local search methodologies, adaptive memory programming, and memetic algorithms [10 -16]. Baldacci et al. [17] provide a useful recent survey paper. As the literature documents, there is substantial room for improvement in the efficiency and effectiveness of methods for solving large scale instances.

2. Reactive Variable Neighborhood Tabu Search

A successful Tabu Search–based methodology for solving the HFVRPTW is developed by Paraskevopoulos et al. [18] that employs a two-phase solution approach. In the first phase several initial solutions are produced using a semi-parallel construction, compiled with a route elimination procedure to improve the vehicle utilization and reduce, if necessary, the number of vehicle routes. The latter is achieved by an enhanced ejection-chain heuristic, originally proposed by Glover [19]. In the second phase, the objective is to minimize the total distribution cost using a Reactive Variable Neighborhood Tabu Search (ReVNTS) method on a subset of high quality solutions, obtained by the first phase. The basic intuition behind the proposed algorithm is driven by the systematic neighborhood change and the shaking mechanism of the basic VNS scheme [20], coupled with a TS tuned for intensification local search. Furthermore, a specialized solution reformation mechanism is utilized for diversification and sampling of the solution space driven by the accumulated experience acquired during the search.

The performance of ReVNTS was evaluated by conducting multiple computational experiments on benchmark instances derived from Solomon [21] consisting of 100 customer instances R, C and RC, originally proposed for the VRPTW. Vehicle capacities and fixed costs were defined by three different subclasses (a, b and c) of problem sets, proposed by Liu and Shen [8]. The computational results indicated that the proposed semi-parallel construction heuristic yielded solutions of respectable quality within short computational times, while the proposed ReVNTS consistently produces solutions of exceedingly high quality for the HFVRPTW. Notably, the ReVNTS performance on data sets of subclass a and all subclasses of R2, C2 and RC2, achieved a remarkable reduction in the total distribution costs.

3. Concluding Remarks

There are strong connections between the Tabu Search strategy and multiple neighborhood search approaches, such as VNS. The ReVNTS takes advantage of these connections by using TS to perform the local search for given neighborhood structures, while VNS selects the neighborhoods and the starting solutions within these neighborhoods by the randomized shaking mechanism. Thus, a thorough and systematic examination of the solution space is conducted by utilizing powerful trajectory local search and exploitation of different neighborhood topologies.

An important component of ReVNTS is the reactive solution reformation mechanism. Similar to the VNS shaking mechanism, the main idea behind solution reformation is to perform a partial solution reconstruction in a reactive fashion during the search. The reformation mechanism competes to modify the fleet mix structure, while maintaining to the extent possible the sequence of customers served by vehicles. As such it ensures that desirable characteristics of a solution are maintained, while the reformed solution may belong to the basins of attraction of different local optima. Therefore, the information gathered during the search (sequence of customers served by particular vehicle types) is exploited, while the search is both diversified towards promising regions and better sampled close to local optimum solutions.

A promising line of research is to study the advantages, connections and links between TS and multiple neighborhood search approaches to a wider area of discrete combinatorial optimization problems. 4. References

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