Part I Introduction
8.4 Other Variants of VRP
8.4.1 Open VRP
One of the variants of VRP is the OVRP. The important feature of the OVRP is that the vehicles are not obliged to return to the depot.
This kind of problem appears for the companies that do not own a vehicle fleet at all or a vehicle fleet that is inadequate to the demand of all customers.
Therefore, the company is obliged to contract all or part of the product distribution to external couriers. The hired vehicles will be assigned to routes and do not have to return to the company’s distribution center (depot).
The problem solution will provide the company with the minimum number of vehicles that must be hired in order to serve the customers and the set of routes that minimizes the traveling cost. Furthermore, in the situation in which the com- pany has its own vehicle fleet and customer demand varies significantly over time, the solution will provide the proper combination of owned and hired vehicles [12].
A practical example of this kind of problem is the delivery and collection of mail in which, after delivery, the vehicles start collecting new mail and return it to the sort- ing office. Air-courier companies also have to determine such routes for fast and efficient service.
The elimination of vehicle return to the depot, which is a constraint in the VRP, does not actually lead to an easier and less complex problem in the OVRP [13], yet OVRP remains NP-hard. Because of its resolution, it is necessary to find the best Hamiltonian path for each set of customers assigned to a vehicle [14]. Therefore, any exact algorithm for solving the OVRP will certainly have the inefficiency of the exact algorithms for the VRP.
The OVRP received very little attention from the early 1980s to the late 1990s. However, since 2000, several researches have used Tabu search, deter- ministic annealing, and large neighborhood search to solve the OVRP with some success [12].
Li et al. [15] provided an extensive review of the literature on OVRP until 2007.
Other studies in this area are reported in Table 8.1.
8.4.2 Multidepot VRP
In classical VRP, there is only one depot and all vehicles start and end their routes in that depot. In the MDVRP, more than one depot exists. In this problem, every customer is visited by a vehicle based at one of the several depots.
The MDVRP can be viewed as a clustering problem, in the sense that the output is a set of vehicle schedules clustered by depot. Therefore, the MDVRP can be solved in two stages: first, customers must be allocated (assigned) to depots; second, customers assigned to the same depot must be linked together through routes. Ideally, it is more efficient to deal with the two steps simultaneously. When faced with larger problems, however, a reasonable approach would be to divide the problem into as many subpro- blems as there are depots and to solve each subproblem separately [21].
Crevier et al. [22] summarized the works on VRP with multiple depots from 1969 to 2002. Other papers since that period are listed in Table 8.2.
MDVRP with interdepot routes is an extension of the MDVRP in which depots can act as intermediate replenishment facilities along the route of a vehicle. This problem is referred to as the VRP with intermediate facilities (VRP-IF). In a distri- bution system, these facilities are warehouses; in a collection system, these facili- ties represent the sites where the vehicles are unloaded.
There are two common points among these proposed methodologies. First, the MDVRP was decomposed; second, the subproblems were solved sequentially and iteratively.
8.4.3 Mix Fleet VRP
MFVRP is a different kind of VRP that differs from the classical one in that it deals with a heterogeneous fleet of vehicles having various capacities and fixed and variable costs. The routing cost is the sum of the fixed and variable cost wherein the variable cost is in proportion to the travel distance.
Table 8.1 OVRP Literature
Author(s) Year Type Algorithm
Repoussis et al. [16] 2007 OVRPTW
Letchford et al. [17] 2007 COVRP Branch-and-cut algorithm
Aksen et al. [18] 2007 COVRPTW Modified ClarkeWright parallel savings algorithm, a nearest insertion
algorithm, and a Tabu search heuristic Fleszar et al. [19] 2008 OVRPTW,
DCOVRP
Variable neighborhood search algorithm Derigs and Reuter [20] 2008 OVRP Attribute-based hill climber
COVRP, capacitated OVRP; OVRPTW, OVRP with time window; COVRPTW, capacitated OVRP with time window;
DCOVRP, distance-constrained OVRP.
There are three kinds of MFVRP in the literature. Introduced by Golden et al. in 1984 [34], the first one uses the same value for the variable costs regardless of the vehicle type and has an unlimited number of vehicles of each type. It is regarded as the vehicle fleet mix (VFM), the fleet size and mixed VRP, and the fleet size and composition VRP.
The second type, proposed by Salhi et al. [31], considers different variable costs dependent on the vehicle type and also has an unlimited number of vehicles of each type. It is the heterogeneous VRP (HVRP), VFM with variable unit running costs, and MFVRP.
Taillard [32] introduced the last type that differs from the second type in which there are restrictions on the number of available different vehicles of each type.
Because of the complexity of the MFVRP, some attempts have been made to formulate it using mixed-integer linear programming, but no exact algorithm for MFVRP has ever been developed [33].
Here, the literature of MFVRP is categorized in Table 8.3.
8.4.4 Split-Delivery VRP
SDVRP is a relaxation of the VRP in which the same customer can be served by different vehicles if it reduces overall costs. This problem was first introduced by
Table 8.2 MDVRP Literature
Author(s) Year Type Algorithm
Giosa et al. [21] 2002 MDVRPTW Designing six heuristics for assigning customers to depots and the same VRP heuristic for each depot Angelelli and Speranza [23] 2002 VRP-IF A Tabu search-based algorithm Wasner and Zapfel [24] 2004 MDLRP for
planning parcel service
Heuristic based on local search
Polacek et al. [25] 2004 MDVRPTW Variable neighborhood search Nagi and Salhi [26] 2005 VRPPD,
MDVRPPD
Several heuristic methods for VRPPD can be modified to tackle
MDVRPPD
Ho et al. [27] 2008 MDVRP Hybrid genetic algorithm
Crevier et al. [22] 2007 MDVRP with interdepot routes
Heuristic combining adaptive memory principle, Tabu search for solution of subproblems, and integer programming Chunyu and Xiaobo [28] 2009 MDVRPB Hybrid genetic algorithm Yu et al. [29] 2010 MDVRP Ant colony metaheuristic Sombuntham and
Kachitvichayanukul [30]
2010 MDVRPPD in time window
Particle swarm optimization algorithm
MDVRP, multidepot VRP; MDVRPPD, MDVRP with pickup and delivery; MDVRPTW, MDVRP with time window;
MDLRP, multidepot location routing problem; MDVRPB, multidepot VRP with backhauls.
Dror and Trudeau [52]. They proved that split deliveries result in savings, both in the total distance traveled and the number of vehicles utilized.
An example in Gendreau et al. [53] shows that when the number of demand points goes to 1N, the ratio of the optimal value of the SDVRP over that of the corresponding CVRP approaches 1/2. These savings are more significant when
Table 8.3 MFVRP Literature
Author(s) Year Type Algorithm
Golden et al. [34] 1984 FSMVRP Saving heuristic based on
ClarkeWright method, two-step procedures
Gheysens et al. [35] 1984 FSMVRP Penalty function approach Gheysens et al. [36] 1986 FSMVRP Two-stage method
Salhi et al. [31] 1992 VFM Route perturbation (RPERT) procedure for different variable costs
Salhi and Rand [37] 1993 FSMVRP Extension of RPERT procedure Rochat and Semet [38] 1994 Heterogeneous
fixed fleet
Tabu search
Osman and Salhi [39] 1996 VFM Modified version of RPERT, called MRPERT, allowing search process to restart several times to produce several solutions
Taillard [32] 1999 VFM Heuristic column-generation method Liu and Shen [40] 1999 FSMVRPTW Insertion-based savings heuristics Renaud and Boctor [41] 2002 FSMVRP Sweep-based algorithm
Wassan and Osman [33] 2002 FSMVRPTW New variants of Tabu search mixed with reactive Tabu search concepts, variable neighborhoods, special data-memory structures, and hashing functions
Dullaert et al. [42] 2002 MFVRPTW Sequential insertion heuristic for the FM Tarantilis et al. [43] 2004 HFM Backtracking adaptative threshold
accepting
Dell’Amico et al. [44] 2006 FSMVRPTW Constructive insertion heuristic and metaheuristics algorithm Belfiore and Favero [45] 2007 FSMVRPTW Scatter search approach
Braysy et al. [46] 2008 FSMVRPTW Multirestart deterministic annealing metaheuristic
Lee et al. [47] 2008 VFM Tabu search and set partitioning Brandao [48] 2009 VFM Deterministic Tabu search algorithm Braysy et al. [49] 2009 FSMVRPTW Three-phase metaheuristic
Liu et al. [50] 2009 FSMVRP Genetic algorithm-based heuristic Repoussis and Tarantilis [51] 2009 FSMVRP Adaptive memory programming
solution approach
FSMVRP, fleet size and mixed VRP; VFM, vehicle fleet mix; FSMVRPTW, FSMVRP with time window.
average customer demand is more than 10% of the vehicle capacity [54]. Ho and Haugland [55] gave a survey on SDVRP up to 2002. Other works since then are reported in Table 8.4.
8.4.5 Periodic VRP
In the PVRP, a set of customers has to be visited on a given time horizon one or more times. Different customers usually require different numbers of visits in that certain time horizon. PVRP with service choice (PVRP-SC) is a variant of the PVRP in which the visit frequency to nodes is a decision variable of the model.
This can result in more efficient vehicle tours or greater service benefit to customers [70].
Solving the problem requires assigning a visiting schedule to each customer. For each day of the time horizon, the routes of the vehicles must be defined in such a way that all customers whose assigned schedules include that day are served.
Table 8.4 SDVRP Literature
Author(s) Year Type Algorithm
Song et al. [56] 2002 SDVRP
(newspaper logistic problem)
Ho et al. [55] 2004 VRPSDTW Tabu search
Archetti et al. [57] 2006 SDVRP Tabu search Gendreau et al. [58] 2006 SDVRTW
Lee et al. [59] 2006 SDVRP (split pickups)
Exact algorithm based on shortest path
Campos et al. [60] 2007 SDVRP Scatter search
Chen et al. [61] 2007 SDVRP Heuristic combining mixed-integer program and record-to-record travel algorithm
Nakao and Nagamochi [62] 2007 SDVRP Dynamic programming
Archetti et al. [63] 2008 SDVRP Method based on Tabu search and using integer programming Suthikarnnarunai [64] 2008 SDVFM Sweep heuristic method Derings et al. [65] 2009 SDVRP Local search-based metaheuristic
Bolduc et al. [66] 2010 SDVRP Tabu search
Aleman et al. [67] 2010 SDVRP Adaptive memory algorithm Moreno et al. [68] 2010 SDVRP Proposing an algorithm to obtain
lower bounds
Gulczynski et al. [69] 2010 SDVRP-MDA Heuristic applying modified ClarkeWright saving algorithm
SDVRPTW, SDVRP with time window; SDVFM, split-delivery vehicle fleet mix; SDVRP-MDA, SDVRP with minimum delivery amounts.
The split-pickup VRP is the same as SDVRP, but products have to be picked up from suppliers.
Therefore, a VRP has to be solved for each day of the planning horizon. In such a case, the choice of the visiting schedules and the definition of the routes are two interrelated problems. This feature is essential in some applications such as waste- collection problems in which each customer has to be served in a given period (e.g., twice a week).
Published papers on this subject from 1974 to 2005 are reported in Hemmelmayr et al. [71], and the summary of other works since then are reviewed briefly in Table 8.5.
8.4.6 Stochastic VRP
The deterministic CVRP has been widely studied in the literature. In the classic definition of VRP, the associated parameters such as cost, customer demands, and vehicle travel times are deterministic. SVRPs arise when some elements of the problem are random.
Common types are the following:
G In the VRP with stochastic demands (VRPSD), each customer’s demand is assumed to follow a given probability distribution instead of having a single known value. The actual customer demand is known only on arrival at the customer’s location.
G In the VRP with stochastic travel time, the matrix of travel time is not deterministic.
G In the VRP with stochastic customers, the set of customers is not known with certainty and each customer has a probability of being present.
G In the VRP with stochastic service time, the service time of each customer is not deterministic.
Table 8.5 PVRP Literature
Author(s) Year Type Algorithm
Delgado et al. [72] 2005 Minimizing labor requirements in a PVRP
Two-level heuristic
Francis and Smilowitz [70] 2005 PVRP-SC Continuous approximation model Belanger et al. [73] 2006 PVRPTW Nonlinear integer multicommodity
network flow formulation and new branch-and-bound strategies in branch and price
Alegre et al. [74] 2007 PVRP (pickups) Scatter search Alonso et al. [75] 2008 Site-dependent
multitrip PVRP
Tabu-based algorithm Pirkwieser and Raidl [76] 2008 PVRPTW Variable neighborhood search Pirkwieser and Raidl [77] 2009 PVRPTW Column-generation method Pirkwieser and Raidl [78] 2009 PVRPTW Integer linear programming solver
with variable neighborhood search
MDPVRP, multidepot PVRP.
In site-dependent multitrip PVRP, a vehicle can have multiple trips during the day and they are site-dependent—i.e., not every vehicle can visit every customer.
SVRPs differ from their deterministic ones in several aspects. Solution method- ologies are more intricate and combine the characteristics of stochastic and integer programs. SVRPs are often computationally intractable; therefore, only relatively small instances can be solved to optimality, and good heuristics are hard to design and assess [79]. SVRPs can be cast within the framework of stochastic programming.
SVRPs are usually modeled using mixed- or pure-integer stochastic programs or as Markov decision processes. All known exact algorithms belong to the first category.
Tillman in 1969 [80] addressed the CVRPSD for the first time. He considered a multidepot variant of the CVRP with Poisson-distributed demands. The model con- sidered a cost trade-off between exceeding the vehicle capacity and finishing the route with excess capacity [81].
The published papers covering all kinds of SVRP are categorized in Table 8.6 by specifying the parameters that have been assumed stochastic.
8.4.7 Fuzzy VRP
There is widespread evidence that the exact values of the mean demands, travel times, numbers and locations of customers, and so on that follow probability distri- butions are very difficult to obtain. In some new systems, it is also hard to describe the parameters of the problem as random variables because of insufficient data to analyze the distribution. Using methods from fuzzy sets theory makes it possible to successfully model problems that contain an element of uncertainty, subjectivity, ambiguity, and vagueness.
Fuzzy logic was used by Teodorovic and Pavkovic [130] in VRP when the demands were uncertain. The model was based on the heuristic sweeping algo- rithm, rules of fuzzy arithmetic, and fuzzy logic.
Cheng and Gen [131] introduced the concept of fuzzy die-time in the vehicle- routing and scheduling context. They represented the fuzzy time window in two types: the tolerable interval of service time and the desirable time for service.
Usual approaches consider the tolerable interval of service time without minding customers’ desired time. Their fuzzy approach can handle both kinds of customers’
preferences simultaneously. Table 8.7 categorizes the papers on VRPF.