ApplicationsA multi-compartment vehicle routing problem arising in the collection of olive oil in Tunisia☆
Introduction
In this paper, we introduce, model and solve a real-world application of a multi-product, multi-period and multi-compartment vehicle routing problem (MPPC-VRP) arising in the collection of olive oil in Tunisia. In 2012, that country was the fourth largest exporter of olive oil worldwide, with an export production of 163,000 tons. This amount was expected to increase in 2013 according to the General Directorate for Research at the Ministry of Agriculture. For climatic and geographical reasons, olive groves are rather widespread in the central part of the country, as shown in Fig. 1. Collecting olive oil is particularly important during the four-month production season. It mobilizes considerable human and material resources, and timeliness is crucial in this operation. The producers work non-stop 24 h a day in order not to damage the harvest. On any given day, olive oil collection is carried out over six periods lasting almost 24 h in total. This activity is performed by a fleet of capacitated heterogeneous vehicles, with compartments of equal or different sizes, all equipped with a debit meter, enabling the decision maker to have full knowledge of the load contained in each compartment at all times. The oil must be collected before the producer runs out of storage space. A good forecast is available for the production rate of each product by each producer.
Olive oil comes in three different grades known as extra, virgin, and lampante. The top two grades with superior tastes are extra and virgin, which are suitable for consumption, whereas lampante oil is mostly destined for industrial uses. The transportation is regulated by law to protect the natural flavors of the oils. In particular, at each producer site, a quality controller is in charge of checking the oil grade proposed. Once the quality control process has been completed with success, the quality controller seals the tank containing this offer. Thereafter, once the vehicle loading starts, it cannot be stopped until the tank is empty. In addition, the different grades must be kept separate during transportation, hence the need to have multi-compartment vehicles. It is forbidden to load superior grades immediately after lampante oil in the same compartment, unless it has been cleaned before the changeover. The cleaning activity incurs a cost and takes time.
Routing problems with a cleaning activity have not been widely studied from a scientific perspective, but similar constraints appear in other contexts. Oppen et al. [33] consider the problem of transporting different types of live animals from farms to slaughterhouses by means of multi-compartment vehicles. They add time between consecutive tours to allow for unloading and disinfection of the vehicles. Hvattum et al. [24] deal with a tank allocation problem arising in the shipping of bulk oil and chemical products by tanker ships. They consider that a cleaning activity is required if two incompatible products are assigned to the same compartment within less than three trips.
The use of fleet with several compartments is common in fuel and oil distribution [2], [8], [10], [11], [15], [38], [42] and in some maritime applications [5], [22], [24], [40]. Transporting oil and fuel with multi-compartment vehicles is more challenging and interesting from a scientific point of view than transporting food, where dry, refrigerated and frozen commodities can be pre-assigned to suitable compartments. In this case, the loading problem reduces to a simple capacity checking procedure [13], [14], [30], [31]. In contrast, in fuel transportation, a routing problem and a compartment assignment problem must be solved jointly. For more details on the loading aspect arising in vehicle routing problems with compartments, see Lahyani et al. [29] and Pirkwieser et al. [36]. Multi-compartment problems have also been investigated in the context of inventory-routing [7], [33], [35], [43]. Other multi-period researches dealing with the joint optimization of transportation and inventory include [1], [3], [23].
Multi-compartment vehicles are often not equipped with debit meters, which implies that a compartment must satisfy at most one request. This scenario has been extensively studied [8], [9], [10], [11], [37], [43]. Coelho and Laporte [6] introduce and define the generalized case where compartments may be equipped with debit meters, and the load of a compartment may be split between different customers. They also distinguish between the cases where customers may or may not be visited by more than one vehicle per period. The authors assess the difficulty of the problem with split compartments and multiple visits per period. Derigs et al. [13] provide a literature review on the VRP with compartments as it arises in food and petrol delivery.
In this paper, we introduce other complicating constraints reflecting the complexities of the olive oil industry. The mathematical model developed for this application considers two vehicle types which differ in the number and sizes of the compartments. In the olive oil industry, a producer typically offers several products in different quantities in each period. The grades and quantities offered depend on the producer. All offers must be picked up in each period since drivers are hired per period. These assumptions, together with the presence of the cleaning activity, increase the difficulty of the problem and make it almost impossible to solve by exact algorithms for most instances of realistic sizes.
From a scientific perspective, we introduce, model, and solve exactly a difficult and rich variation of the well-known vehicle routing problem [28]. Part of the complexity of the problem arises from the requirement to clean the compartments, depending on the assignment and scheduling decisions. From a practical standpoint, we provide a tool that can assist managerial decision making at the tactical and operational levels. In particular, we are able to compare several transportation alternatives for the service provider and to evaluate the potential routing savings yielded by replacing single compartment vehicles with multi-compartment ones. Moreover, we show how our model can be extended to a number of other industries operating under different assumptions.
The remainder of this paper is organized as follows. In Section 2 we provide a formal description of the problem and we introduce a mathematical model complemented with known and new valid inequalities. The branch-and-cut algorithm is described in Section 3. We present computational results on real and artificially generated instances in Section 4, followed by conclusions in Section 5.
Section snippets
Mathematical description of the problem
We first introduce some notation, followed by a mathematical model and valid inequalities.
Branch-and-cut algorithm
We have implemented a branch-and-cut algorithm capable of solving the formulation just presented. All variables of the formulation are explicitly handled by the algorithm. Since the number of constraints (16), (17), (18), (19), (20), (21), (22), (23), (24) is polynomial, they are added a priori to the model. In the sequel, we will show how each subset of inequalities impacts its solution. In contrast, we cannot generate all subtour elimination constraints (8) a priori since their number is
Computational experiments
In this section we describe the computational experiments carried out in order to assess the performance of our model and algorithm. We provide in Section 4.1 details of the real instances we have obtained from our Tunisian partner, and the instances generated with a different fleet composition. In Section 4.2 we describe the results of computational experiments performed to evaluate the effectiveness of the cuts and valid inequalities, and we compare our solutions with those corresponding to
Conclusions
We have tackled a real-world and rich multi-compartment vehicle routing problem arising in the olive oil collection industry. This practical application concerns the pick-up of one or more commodities from a set of geographically scattered producers in the center of Tunisia. We have presented a mathematical model including known and new valid inequalities, as well as a branch-and-cut algorithm for its solution. We have shown how our model can be adapted to cover a variety of other industries
Acknowledgments
This work was partially supported by the International Campus on Safety and Intermodality in Transportation, the Nord–Pas-de-Calais Region, the European Community, the Regional Delegation for Research and Technology, the French Ministry of Higher Education and Research and the National Center for Scientific Research. The internship of the first author in CIRRELT was partly supported by the University Lille Nord de France and Fondation Centrale Initiatives. This work was partly supported by the
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This manuscript was processed by Associate Editor Bish.