2. The Vehicle Routing Problem: Models and Solution Methods
2.5. Examples of Heuristic Methods
This section briefly introduces some of the most common heuristic methods used for finding optimal or near optimal solutions for small size problems, or for improving (local search methods) on initial solutions found via initial solution construction methods. We note that classical improvement heuristics have two properties; these are
30 that the solution never deteriorates and always remains feasible. The quality of a heuristic method is assessed on four criteria namely speed, flexibility, accuracy and simplicity (Cordeau et al., 2001). Note that in the following sub-sections we are going to describe only those construction and improvement heuristic methods that are directly used in this study. For others, relevant references are provided in sub-section 2.5.1.2.
2.5.1. Construction-based Heuristics
Construction-based heuristics are heuristic methods that create an initial solution from raw data. With small problems, it is possible to find the optimal solution and hence not necessarily have to go through an improvement stage. However, as the problem size becomes larger, these construction-based heuristics could only provide a reasonable initial solution, thus requiring other improvement heuristics to improve the quality of solution.
In the following parts, a brief description of the most well-known construction-based heuristics is introduced with a brief assessment of the quality of the heuristic.
2.5.1.1. The Sweep Algorithm
The sweep algorithm was first introduced by Gillett and Miller (1974). It first selects a starting customer, shoots a “beam” from the depot to the starting customer and rotates clockwise or counterclockwise adding customers one-by-one to form a tour. If the capacity constraint is violated, a new route is initiated until all customers have been assigned. Figure 2.3 shows the visual representation in a clock-wise format.
Like the Clarke-Wright savings and the NNH, the sweep algorithm has a relatively high speed and simplicity. The accuracy is only mediocre and it is relatively inflexible. For
31 more information and the various ways it has been used to the VRP and its variants, readers are referred to Gillet and Miller (1974), Laporte et al. (2000), Renaud and Boctor (2002), Salhi, Wassan and Hajarat (2013). We shall elaborate more on the sweep methodology in the later chapters of this thesis since it is adopted in our algorithm implementations.
Figure 2.3: A visual representation of the sweep procedure
2.5.1.2. Other Construction-based Heuristic for VRPs
There are several other construction-based heuristics methods that are successfully used for VRPs. Among these some popular methods are; the savings algorithm proposed by Clark-Wright(1964), for more information on the savings algorithm and its enhancements, readers are referred to Laporte et al. (2000), Toth and Vigo (2002), Altinel and Oncan, (2005). The nearest neighbour heuristic (NNH) method, for more
Depot
32 details of Nearest-Neighbour heuristic and its applications on VRPs, readers are referred to Rosenkrantz et al. (1977), Golden et al. (1980), Solomon (1987), Fisher (1994), Ganesh et al., (2007). The two-phase methods (e.g., cluster-first, route-second method introduced by Fisher and Jaikumar (1981), and the route-first, cluster-second method introduced by Beasley (1983)). For more details about these methods and their different types/extensions, see Christofides et al. (1979), Renaud et al. (1996) and Toth and Vigo (2002). The insertion method introduced by Flood (1956); for details, see Salhi and Sari (1999). The GENI (Genius) heuristic proposed by Gendreau et al. (1992), for more details on this method, see Gendreau et al. (1994).
2.5.2. Intra- and Inter-Route Improvement Heuristics
Intra- and/or inter-route improvement heuristics are used for improving initial solutions generated from construction heuristics. There are various common improvement heuristics that have been investigated and we shall briefly introduce those which are directly used in this study in the following sub-sections and references are provided for other types of improvement heuristics in sub-section 2.5.2.3.
2.5.2.1. Transfer Heuristics
Transfer heuristics work by removing customer i from its initial position of route I and reinserting it into a different position in the same route I for intra-route optimisation.
Whereas, for inter-route optimisation, customer i is removed from its initial position of route I and reinserting it into a new position in route J. Provided, the transfer of customer i results in an overall cost minimization without violating vehicle capacity constraints, the solution routes are updated. Two of the more commonly used transfer heuristics are briefly introduced in the following sections.
33 2.5.2.2. Swap Heuristics
Swap heuristics swap customers i and j in route I for the intra-route optimisation case, and; customer i from route I with customer j from route J for the inter-route optimisation case. The new solution resulted from the swap is kept if the new solution reduces the overall cost without violating vehicle capacities constraints (Waters, 1987).
This can be extended for swapping several customers, as seen in the following sections.
2.5.2.3. Other Improvement Heuristics for VRPs
In the VRP literature there exist several other improvements/local search heuristics that have been used to improve the solution of VRP and its various extensions. Some of these local search heuristics are: Or-Optimisation introduced by Ilhan Or (1976), r- Optimisation proposed by Croes (1958), see Bock (1958) and Renaud et al. (1996) for different extension of r-optimisation. The -interchange (lambda-interchange) introduced by Osman (1991, 1993). Composite Move Heuristics also known as the cyclic transfers algorithms introduced by Thompson and Orlin (1989). One of the most promising cyclic transfers algorithms, the Ejection Chain Process was proposed by Glover (1991, 1992), see Thompson and Psaraftis (1993), and Rego and Roucairol (1996) for its VRP implementations.