The general LPDPTW has the following specifications: vehicles (transportation assets), depots (home location of vehicles), supply facilities, requests, routes, and time considerations (both delivery and pickup). Natural extensions to the LPDPTW include: the number of trips per vehicle, the number of visits per customer (traditionally held at one in the literature), hubs (locations for consolidation and transshipment), and commodities (traditionally held at single commodity).
4.1.1 LPDP Hierarchy Development
This research characterizes the LPDP and TDP in the classification scheme of Barnes and Carlton (1996). Carlton’s (1996) work provided a multi-tiered framework that characterized the GVRP hierarchy and represents an extension of Bodin et al. (1983) vehicle routing and scheduling classification. Carlton’s (1995) hierarchy represents the TSP, VRP and PDP class problems as succeeding levels. The TSP and VRP levels are connected in the presence of vehicle capacity constraints and the VRP and PDP levels are
connected if precedence and coupling constraints exist. Each level includes the common characteristics of the GVRP including:
1. Number of vehicles - single vehicle (SV)
- multiple vehicles (MV)
2. Type of vehicles - homogeneous vehicles (H)
- non-homogeneous vehicles ( H )
3. Number of depots - single depot (SD)
- multiple depots (MD)
4. Route Length (RL) constraints – distance or time a vehicle may travel 5. Time Windows (TW) – service must occur within a specified time window The TDP, under Carlton’s classification hierarchy, is partially classified as a MVH , MD, PDP, with RL and TW constraints. Crino et al. (2004) extends Carlton’s (1995) classification with the addition of the following four characteristics:
1. Trips per vehicle - single trip (ST)
- multiple trips (MT)
2. Services per customer - single service (SS)
- multiple services (MS)
3. Types of commodity - single commodity (SC)
- multiple commodities (MC)
4. Existence of transshipment points (T)
These four additions are typically assumed to be single or non-existent in the case of transshipment points in the TSP, VRP and PDP class problems but are necessary
to capture the true nature of the TDP. A TDP vehicle is expected to make multiple trips during the course of any given operation. This is especially true in any effort to reduce the logistics footprint so multiple trips is a needed characteristic of the TDP. Logistics requirements for a unit in the theater typically exceed the capacity of a single vehicle and require multiple services to complete the delivery of all requirements to the unit. Any given TDP scenario is characterized by a limited number of APOD/SPOD locations that all logistics flow through. This restriction requires multiple visits to a supply point to meet the demands of the problem. These situations make multiple services a characteristic of the TDP. TDP logistics requirements are multiple commodity requests that either have different vehicle fill efficiencies or require different vehicle types. For example, a Class III (Fuel) demand requires an entirely different vehicle type (tanker) than a Class I (Food) demand (flatbed trailer). Additionally, a Class V (Ammunition) demand generally exceeds a vehicle’s weight limit before its volume limit is exceeded. Conversely, Class I demands generally exceed the vehicle’s volume limit. This multi- commodity aspect impacts the fleet mix assigned to each depot. In addition, the required travel distances between APOD/SPOD and customers may exceed a vehicle’s allowed travel limit. This shortcoming may require the establishment of a transshipment point in the theater to serve as a temporary storage point prior to additional forward movement of supplies to a customer. This transshipment point becomes both a pickup and delivery point. The above four conditions represent Crino’s et al. (2004) four additional characteristics and are included as part of the dissertation solved in this TDP. The TDP, under Carlton’s and Crino’s classification hierarchy, is then partially classified as a
MVH , MD, PDP, RL, TW, MT, MS, MC, T problem. However, this classification does not capture the TDP’s requirement of selecting the required depots and supply facilities and assigning vehicles to the depot for the distribution network.
The classification schemes of Carlton (1995) and Crino et al. (2004) make the assumption that the set of depots are given and fixed. The TDP’s objective is to minimize the distribution footprint of the theater and therefore selects, based on cost, the required depots and supply facilities from an available set of potential sites. This cost based selection adds a new location (L) characteristic to the problem description hierarchy. The TDP is classified as a traditional PDP if it has a given fixed depot list (FDL) and becomes the LPDP if it contains a set depot list (SDL). The set depot list implies that the distribution network consists of a selected subset of the list. Figure 4-1 provides a representation of the LPDP classification hierarchy.
The LPDP hierarchy (Figure 4-1) displays 32 characteristic description combinations for the physical network of the TDP. The first column (level 5) is the basic PDPTW, with standard assumptions that Carlton (1995) developed in his research effort. The hierarchy moves from left to right becoming more general as Crino’s (2004) additional constraints are added with the requirement of locating the assigned depots, supply facilities and allocating vehicles to the depot. The TDP is now classified under the LPDP as a MVH , MD, LPDP - SDL, RL, TW, MT, MS, MC, T problem.
Specific TDP scenarios contain several additional characteristics not described by the above LPDP hierarchy. Constraints at a depot or transshipment point may impact a
Figure IV-1 LPDPTW Hierarchy for the TDP Ba s ic PD P T W L e v e l 5 L ev e l 6 L ev e l 7 L ev e l 8 L ev e l 9 L e ve l 10 MCMC LP DP T W
vehicle’s route timing. These constraints include such items as on-hand commodity availability or throughput capacity. A vehicle may be forced to wait at a node for supplies to become available (delivered from another location) or for throughput capacity to become available for servicing. A typical throughput constraint is the maximum on the ground (MOG) ability of any given airfield or the number of container spaces in a port. This constraint limits the number of vehicles that can be processed at any given time and therefore limits the amount of supplies that can process through the point. These constraints place time restrictions on vehicles and impact on when they travel along routes. This research refers to these restrictions as timing constraints (TC) to separate them from time window and route constraints.
This section has presented a characterization of the TDP solved in this research. The next section provides a formal presentation of the mathematical formulation of the LPDPTW.