A comparison of a Metaheuristic with an Exact Method to solve the Vehicle Routing Problem: Application to a Case Study in the Distribution Industry André Guerreiro Instituto Superior Técnico, Universidade Técnica de Lisboa, Lisboa, Portugal Abstract In this paper, it is evaluated a vehicle routing problem model applied to a real case of a distribution company. A combinatorial optimization method was implemented based on a rule‐guided heuristic – a tabu search based metaheuristic – in order to obtain practical results and to compare them with the real plan used daily by the company. The problem’s extension is referred in the literature as Heterogeneous Fleet Vehicle Routing Problem with Time Windows and Split Deliveries (HFVRPTWSD). The objective is to minimize the total travelled distance, the necessary number of vehicles and the sum of travel time through the establishment of a set of routes between all clients and a central depot for the given time horizon. The main purpose is to satisfy all clients’ requests while complying with a visit schedule. A number of constraints define the problem working environment, with special attention to time windows constraints, truck capacity and accessibility constraints. A relaxation of the original problem – split delivery – is allowed whenever it improves the solution. The output provided by an implementation of a VRP‐solving algorithm will be compared with the results obtained from an exact method applied to a smaller part of the problem. After comparison with the global optimum, we may capture the algorithm’s power and therefore apply it to the overall problem. The case study analyzed is based on a Portuguese company that distributes groceries from distribution centers to urban retailers in Portugal. 1. Introduction The present work was motivated by the need of obtaining In this paper, it is presented a vehicle routing problem satisfactory results in a short amount of time. (VRP) applied to a real Case Study in the distribution area. Consequently, it was developed an approximate method to address those requirements: a metaheuristic inspired on The analyzed problem presents several specifications that Tabu Search and several diversification procedures. add or relax constraints to the original VRP. Therefore, a most suitable variation of the VRP had to be considered, The main goal of this thesis was to develop a model that including Time Windows constraints (VRPTW), Capacity could obtain high‐quality solutions – fully compliant with Constraints with Heterogeneous Fleet (HFVRP) and Split all problem constraints – in a reasonable amount of time in Deliveries (VRPSD) allowance whenever it improves a order to proceed with a successful implementation in the solution. daily operations of the JM company. The defined problem may be described as HFVRPTWSD – A broader objective intends to verify that an approximate Heterogeneous Fleet Vehicle Routing Problem with Time method gets advantage in performance over an exact Windows and Split Deliveries (Belfiore, 2006). method. Furthermore, some additional constraints have to be 2. Problem Definition and Case Study Analysis attended in order to comply with the Case Study problem’s In this paper, it is considered a HFVRPTWSD problem restrictions, such as tour duration and clients’ accessibility (Heterogeneous Fleet Vehicle Routing Problem with Time constraints. Windows and Split Deliveries) consisting of four instances The group Jerónimo Martins (JM) runs five Distribution (4 days of planning) with dimension of 40 clients, each with Centers (DC) in Portugal, three in the Northern region and an associated demand and a specific time window to fulfill. two in the South, to serve roughly four hundred Also, there are accessibility constraints that need to be commercial surfaces spread all over the country, under the respected, which require a heterogeneous fleet to serve brands of Pingo Doce, Feira Nova and Recheio. The Case clients with different accessibility conditions, particularly Study described on this paper focus the planning of the due to urban placement strategy. Another aspect included South Region, more specifically the planning of the DC in the accessibility constraint consists on the unloading located in Azambuja. process. Either the client’s facility includes a docking The problem solved in this paper was previously station or the truck is equipped with an elevator to unload approached by Figueiredo (2007) using an Exact Method. the cargo during each delivery. Therefore, this constraint In his work, Figueiredo (2007) achieved significant was defined as the largest truck (in volume capacity) that improvements when compared with the real planning. can supply each client. However, due to limitations imposed by the exact nature The problem’s time horizon is of one day and client’s time of his model, it was not possible to solve the problem in its windows are divided in morning planning and afternoon full extent in a reasonable amount of time. planning. The distance and time matrixes were collected 1 from real measurements and the resultant average speed The remaining product categories (perishable) have a short is of 40 km/h. Service times are composed by a fixed life time. Hence, the associated demand has low variability, component of 20 minutes per visit plus a variable high frequency and narrow time windows, which results component of 2 minutes per unloaded palette (unit of essentially in fixed routes, with little interest for volume amount). optimization purposes. Due to computational effort limitations, the described Finally, the resolution of the problem allows split deliveries problem, originally formulated by Figueiredo (2007), whenever it improves the quality of the solution during the represents a short example of the real planning. The real search process. Due to a service time increase per client extent of JM’s group distribution operation in Portugal visit (Figueiredo, 2007), in this work the maximum number consists of roughly 400 clients and 5 distribution centers. of visits per client was limited to two when Split Deliveries Also, it does not include some restrictions of the real are allowed. planning, such as a multi‐depot with multi‐product The allowance of Split Deliveries on a VRP, first introduced scheduling. On the other hand, the incurred simplification by Dror e Trudeau (1989), brings greater advantages under excludes mixed cargo routes (with more than one product certain conditions. category) that would certainly improve the quality of the obtained solutions. Some other operations were not According to Archetti et al. (2006), the utilization of Split contemplated in this formulation, such as cross‐docking, Deliveries is justified when the following criteria are met: backhauling and transshipment operations. and demand variability is low. Therefore, the formulated problem describes the planning Graph 1 shows that the use of Split Deliveries in the Case for a single warehouse of the Azambuja Distribution Study problem allows reduction on the number of routes. Center: 5401 – Non Perishable product category (NP). The This comparison is also valid for vehicles with 24‐palette planning associated with this product category is and 31‐palette capacity but in a lesser extent since most characterized by its large variability, medium frequency clients cant’ be served by 31‐palette trucks due to and flexible time windows, resulting in variable routes and accessibility constraints and the average demand of clients therefore subject to optimization. that can be served by 24‐palette trucks is generally less than half of its capacity. Necessary number of routes as a function of clients' average demand 11 45 2,00 40 1,80 35 1,60 30 1,40 routes 1,20 of 25 1,00 20 0,80 r(VRP)/r(VRPSD) 15 0,60 Number 10 0,40 5 0,20 Ratio 0 0,00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Clients Average Demand r(VRP) r(VRPSD) r(VRP)/ r(VRPSD) Graph 1 – Necessary number of routes to supply 40 clients with vehicles of capacity of 21 palettes as a function of clients’ average demand: comparison of VRP and VRPSD scenarios 3. State‐of‐the‐art referred as an extension of the classical Local Search methods, combining LS with short term memory of some In the past two decades, several hundreds of published solution attributes. articles addressed applications of the Tabu Search (TS) in combinatorial optimization problems. The initial concept 3.1 Metaheuristics vs Exact Methods was introduced by Glover (1986), described as a Metaheuristics such as Tabu Search had evolved through metaheuristic inspired on a lower‐level heuristic designed the time heading towards a greater performance. It to address specific problems. A metaheuristic can be allowed the resolution of bigger and more complex defined as an iterative method that guides the search problems and narrowed the gap to Exact Methods. process of the subordinated heuristic in order to efficiently According to Brejon (1998), several aspects reinforce the produce high quality solutions (Voß, 1999). utilization of metaheuristics: Like other Local Search (LS) procedures, TS explores the ‐ The Exact Methods are prohibitive for large problems search space by moving from a current solution S to the due to computational effort requirements; best found solution in its neighborhood N(S). It can be 2 ‐ The approximate methods are simple, easy to The comparison with the real planning pointed out several implement and to use and require less resources, important conclusions about the applicability of an which is particularly useful for a periodic problem like approximate model in large scale real problems: the the daily operation of a real company; solution obtained represented a cost cut of € 974.0161 ‐ The available data is not exact or there is limited annually (the obtained results were projected for an information, making necessary the use of average representative week), which corresponds a approximations that may override the error caused by relative saving of 6.3%.