4. The Multiple Trip Vehicle Routing Problem with Backhauls: Formulation and
4.1. The Multiple Trip Vehicle Routing Problem with Backhauls
The MT-VRPB is created in this thesis by blending the characteristics of two well- studied variants of the VRP, i.e., VRP with Multiple Trips (MT-VRP) and the VRP with Backhauls (VRPB). In the MT-VRP a vehicle may perform several routes (trips) within
66 a given time period; and in the vehicle routing problem with backhauls (VRPB) a vehicle may pick up goods to bring back to the depot after the deliveries are made.
Therefore in the MT-VRPB a vehicle may not only make more than one trip in a given planning period but it can also collect goods in each trip, see Sections 3.1 and 3.3 for the descriptions of the VRPB and the MT-VRP respectively. From the real life applications point of view both the MT-VRP and the VRPB can be even more practical than the classical VRP. In real-life routing applications, vehicles can be used more efficiently;
for instance in VRPB, combining delivery and pickup operations can result in saving companies substantial routing costs. Golden et al. (1985) reported that grocery stores in USA saved $165 million by taking advantage of backhauling in 1982. On the other hand, maximising the usage of vehicles as it is done in the MT-VRP results in saving the number of vehicle required and hence savings in total distribution costs. Therefore, by combining the aspects from these two routing problems, a new version of the VRPs that we believe will help bridge the gap between theoretical academic studies and the reality is created. The statement of the MT-VRPB is as follows.
4.1.1. Description of the MT-VRPB
The MT-VRPB can be described as a VRP problem with the additional possibilities of having vehicles involved in backhauling and multiple trips in a single planning period.
In the MT-VRPB the fleet considered is homogenous, a vehicle (note that a vehicle corresponds to a bin and these two terms are used interchangeably in this study) may perform more than one route (trip) in a single planning period and may serve backhaul (pickup) customers after serving all linehaul (delivery) customers, the fleet is operated from a single depot and the demands of all the customers must be fulfilled; the objective is to minimise the overall cost by reducing the total distance travelled. There are
67 implicit cost savings attached with the number of vehicles used. The details of the MT- VRPB are as follows.
Problem characteristics and conventions:
1. A given set of customers is divided into two subsets, i.e., delivery (linehaul) and pickup (backhaul).
2. A homogenous fleet of vehicles is located at a single depot.
3. A vehicle may perform more than one trip in a single planning period.
4. All delivery customers are served before any pickup ones.
5. Vehicles routes containing only backhauls are not permitted; however linehaul only routes are allowed.
6. Vehicle capacity constraints are enforced.
7. Note - The route length constraint is not imposed at this stage, however the model is flexible to add this constraint if needed.
The MT-VRPB is to design a set of minimum cost schedules in which each customer (LH/BH) is visited exactly once by the routes (originating and terminating at the same depot) included in the schedules.
Figure 4.1 presents a graphical illustration of the MT-VRPB. Three homogenous vehicles are shown serving a given set of customers with known demands. The distance of d or d (where d represents the distance of a respective route) combined with the distance of other three routes cannot be served by the same vehicle in a single planning period T (for example, T could correspond to eight hour working day; i.e., T = 480 minutes for each vehicle); hence two separate vehicles (Vehicle 2 and Vehicle 3) are used to serve these routes. In this study, the terms distance and planning period (travel
68 time) are used interchangeably. However, Vehicle 1 performs two trips in a single planning period, since the total distance of d and d is less than a given planning period time T.
Figure 4.1: An example of the MT-VRPB
4.1.2. Graph theoretical definition of the MT-VRPB
The MT-VRPB can be defined on a graph as follows. Let be an undirected network, where is a set of nodes, correspond to the linehaul (delivery) customers and correspond to the backhaul (pickup) customers. is the set of arcs and associated with arc , there is nonnegative given cost (distance between node i and node j). σode ‘0’ represents the depot where a fleet of identical vehicles is located while the other nodes correspond to L and B customer sets. A non-
E.g; T (time) = 480 minutes (8 hours) Planning period time for each vehicle Distance = Time
Vehicle 1
d = 205 Vehicle 2 d = 330
Vehicle 1
d = 212
Vehicle 3 d = 358
Delivery (Linehaul) Customers Pickup (Backhaul) Customers Vehicle 1
d + d <= T 205 + 212 = 417
dR = total length/distance of route R
69 negative quantity is associated with each (L/B) node . Each vehicle has capacity and maximum driving time .
A travel cost and a travel time are associated with each arc Therefore, a route of a vehicle is a least-cost elementary cycle in that passes through a subset of customers starting and ending at the depot such that the customers visited and their total demand does not exceed the vehicle capacity . A route cost (duration) is equal to the sum of the travel costs (travel times) of the nodes traversed. A vehicle schedule is a subset of routes whose combined duration is equal to or less than the maximum driving time . Hence, the MT-VRPB call for the determination of constructing schedules of least total cost in which each customers is visited exactly once by the routes of the schedules.
In the following section we review briefly the exact methods options for the MT-VRPB.