Accepted Manuscript
A Parallel Variable Neighborhood Search for the Vehicle Routing Problem with Divisible Deliveries and Pickups Olcay POLAT PII: DOI: Reference:
S0305-0548(17)30074-6 10.1016/j.cor.2017.03.009 CAOR 4217
To appear in:
Computers and Operations Research
Received date: Revised date: Accepted date:
4 September 2016 18 March 2017 20 March 2017
Please cite this article as: Olcay POLAT , A Parallel Variable Neighborhood Search for the Vehicle Routing Problem with Divisible Deliveries and Pickups , Computers and Operations Research (2017), doi: 10.1016/j.cor.2017.03.009
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HIGHLIGHTS The vehicle routing problem with divisible delivery and pickup is considered. A parallel approach based on variable neighborhood search is proposed. A number of well-known benchmark problems are solved and compared.
Proposed parallel approach outperforms the existing methods.
New best solutions for 179 of 220 benchmark instances are found.
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A Parallel Variable Neighborhood Search for the Vehicle Routing Problem with Divisible Deliveries and Pickups Dr.-Ing. Olcay POLAT
Pamukkale University,
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Dept. of Industrial Engineering
Faculty of Engineering, Room: 450 20160 Denizli, Turkey
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Tel: +90 258 296 3013 Fax: +90 258 296 3262
E-Mail:
[email protected] -Alt: [email protected]
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Web: opolat.pau.edu.tr
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Abstract
A well-known variant of the vehicle routing problem involves backhauls, where vehicles deliver goods
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from a depot to linehaul customers and pick up goods from backhaul customers to the depot. The vehicle routing problem with divisible deliveries and pickups (VRPDDP) allows vehicles to visit each
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client once or twice for deliveries or pickups. In this study, a very efficient parallel approach based on variable neighborhood search (VNS) is proposed to solve VRPDDP. In this approach, asynchronous cooperation with a centralized information exchange strategy is used for parallelization of the VNS
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approach, called cooperative VNS (CVNS). All available problem sets of VRPDDP have been successfully solved with the CVNS, and the best solutions available in the literature have been significantly improved. Keywords: Vehicle routing problem, divisible deliveries and pickups, parallel computing, variable neighborhood search
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ACCEPTED MANUSCRIPT Introduction
The vehicle routing problem (VRP) aims to design the service network of a fleet of homogeneous vehicles originating and terminating at a depot to a set of customers by minimizing total travel distance [1, 2]. Vehicle routing problem with pickup and delivery (VRPPD) extends the problem by allowing transportation of goods from the depot to customers (linehauls) and from customers to the depot (backhauls). The VRPPD is an important logistics problem with wide applications including local postal service, beverage distribution, container shipping etc. Companies are even more interested in
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opportunities for a reduction in total cost and the number of vehicles in the current competitive conditions. Moreover, increasing demand on the environmentally conscious manufacturing results usage of more recyclable goods that require better concentration on the reverse logistics applications [3].
The VRPPD can be basically subdivided into four subclasses: the VRP with clustered backhauls
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(VRPCB), the VRP with mixed linehauls and backhauls (VRPMB), the VRP with simultaneous pickups and deliveries (VRPSPD) and the VRP with divisible deliveries and pickups (VRPDDP). Figure 1 depicts the basic four subclasses with the same configuration, where the square, circles and triangles represent the central depot, delivery goods and pickup goods, respectively. In the VRPCB and VRPMB subclasses, customers require either deliveries or pickups. In the VRPCB, the vehicles first deliver all
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goods to linehaul customers, then pick up goods from backhaul clients (see Figure 1.a). On the other hand, in the VRPMB, the vehicles can make deliveries or pickups in any sequence (see Figure 1.b). In
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the VRPSPD and VRPDDP subclasses, customers require both deliveries and pickups. In the VRPSPD, the vehicles can visit each customer only once and have to perform both operations at that time (see
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Figure 1.c). Finally, in the VRPDDP, the vehicles may visit each customer twice for deliveries or pickups (see Figure 1.d). As seen in Figure 1.d, the divided operations may be carried out on the same
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route in a different order or be executed by different routes. Detailed classifications and a survey related to pickup and delivery problems are presented by Desaulniers, Desrosiers [4], Berbeglia,
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Cordeau [5], Parragh, Doerner [6] and Wassan and Nagy [3].
In the VRPCB, VRPMB and VRPSPD subclasses, when loads cannot be split, the vehicles are allowed to visit each customer only once for pickups and deliveries. However, a route may not be feasible, even if both the total delivery and pickup loads are below its capacity, due to the fluctuating load on a vehicle [3]. Figure 2 represents alternative routes for VRPSPD and VRPDDP as an example. In this
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ACCEPTED MANUSCRIPT figure, each route represents the visiting sequence of the vehicle, delivery and pickup represents the demands of clients, effect represents the positive or negative load of demand on the vehicle, load represents the current total departure load of the vehicle at each node and distance represents the total travelled distance up to arriving at the current node. The bold numbers in the load row point out those clients where the vehicle is overloaded. Consider Figure 2.a. Although 74 units of total delivery load and 96 units of total pickup load do not violate the vehicle capacity constraint, i.e., 100 units, the route becomes infeasible after visiting customers E and F. The reverse of this route also
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provides an infeasible solution (Figure 2.b). Changing the order of customers visited with an insert move may provide feasible solutions. However, this may increase the total travel distance of the vehicle (Figure 2.c). On the other hand, as in the VRPDDP, allowing a vehicle to visit a customer (e.g.,
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E) twice and split delivery and pickup loads may also make the route feasible (Figure 2.d).
In the VRPDDP, in comparison to the VRPSPD, dividing delivery and pickup loads also have the potential to decrease the number of vehicles needed and the total distance covered by increasing
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the capacity utilization rates of the vehicles [7]. On the other hand, the VRPDDP can also be converted into a VRPMB by creating a pair of dummy customers for deliveries and pickups for each
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customer. There are also some similarities between the split delivery vehicle routing problem (SDVRP) and the VRPDDP. The SDVRP variant allows vehicles to visit linehaul customers more than
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once by splitting delivery loads. On the other hand, in the VRPDDP, individual delivery or pickup loads cannot be split into smaller loads; vehicles are only allowed to make deliveries and pickups
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separately, requiring two visits per customer (one for delivery and another for pickup), or simultaneously by visiting customers once. See Archetti and Speranza [8] and McNabb, Weir [9] for
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detailed discussions on the SDVRP. Divisible pickup and delivery is practical when the customers have one or more of the following characteristics: having large demand, having located near depot location and/or in a cluster [7]. Since it is possible to insert a near depot delivery to the beginning of a route or inserting a near depot pickup to the end of a route without significantly increasing the total route distance, the depot near location is a common characteristic of split customers. Nagy, Wassan [7] also show that the absence of very small demands is another factor of effectiveness on divisible pickup and delivery. Being located in a cluster (densely populated area) has also an indicator for possible splitting since making a detour to serve a split customer results with only a small increase in the total route distance. When
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ACCEPTED MANUSCRIPT the customers have these characteristics, serving customers twice may often reduce total distance and the number of vehicles in VRPDDP compared to VRPSPD [7]. In the literature, several VRP studies focused on divisible deliveries and pickups restricted to the use of a single vehicle as a travelling salesman problem (TSP) variant. Mosheiov [10] addressed the TSP with divisible deliveries and pickups (TSPDDP). The author proved that feasibility may be achieved by reinserting the depot into a different edge on the tour. The paper proposed an alternative insertionbased heuristic approach for a solution of the problem. Gribkovskaia, Halskau [11] presented a lasso
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construction heuristic approach using the sequential nearest-neighbor method for the problem. Gribkovskaia, Halskau sr [12] proposed a mixed-integer linear programming model (MILP) by considering various tour shapes occurring in the TSPDDP, and a tabu search metaheuristic was developed to solve the problem. Later, Gribkovskaia, Laporte [13] extended their model and solution method by considering selected pickups, where all loads have to be delivered, but load pickups are
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optional. Hoff and Løkketangen [14] considered restricted mixing, where the vehicle first only delivers loads until the vehicle is partly unloaded, then makes both deliveries and pickups, and finally picks up loads from the customers first visited.
Anily [15] presented the first study that included divisible pickups and deliveries for a fleet of
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identical capacitated vehicles. However, the authors assumed that all deliveries have to be finished before pickups are started on a route. A circular regional partitioning heuristic was proposed to
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create linehaul and backhaul routes, with these routes merged by an assignment method. The VRPDDP was modeled by Salhi and Nagy [16] and Nagy and Salhi [17]. The authors proposed constructive heuristics based on splitting, merging and inserting customers for the VRPB variants
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along with the VRPDDP. Mitra [18] proposed an MILP model for the VRPDDP and a route construction heuristic for solving small-sized problem instances. Later, Mitra [19] presented an
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alternative MILP formulation for the problem and developed a parallel clustering technique to solve the problem. Recently, Nagy, Wassan [7] discussed the relationship between the VRPDDP and the
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VRPB variants in detail and defined benchmark problem instances for the VRPDDP by modifying the VRPSPD test instances of Salhi and Nagy [16]. The authors proposed alternative heuristic approaches to solve the problem by adapting a reactive tabu search approach developed by Wassan, Wassan [20] for the VRPSPD. In general, parallel computing attempts to efficiently explore different regions of solution space as well as reduce the execution time of the basic algorithm by letting various processors work simultaneously, with the common objective of solving the problem [21]. Please see Crainic and Hail [22], Crainic and Toulouse [23] and Alba, Luque [24] for developments in parallel metaheuristics. There are also some studies which executed parallel metaheuristics to solve the VRP variants [25-28]. 5
ACCEPTED MANUSCRIPT The variable neighborhood search (VNS) approach as a powerful metaheuristic methodology for solving a set of combinatorial optimization problems is also widely used in the VRP variants [29]. Please see Bräysy [30]; Polacek, Hartl [31]; Polat, Günther [32]; Avci and Topaloglu [33] and Polat, Kalayci [34] for efficient executions of the VNS on the VRP variants. To the best of the authors’ knowledge, only Polacek, Benkner [35] proposed a parallel VNS approach to solve a VRP variant. They solved the multi-depot VRP with time windows by providing cooperative and adaptive algorithms for the VNS with different data exchange strategies.
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The contribution of this study to the related literature is in twofold: First, developing a very efficient parallel approach that utilizes asynchronous cooperation with a centralized information exchange strategy in VNS for solution of a class of VRP. Second, providing the new best solutions for a number of benchmark test instances for VRPDDP. The remainder of this study is structured as follows: the solution procedure is explained in Section 2; detailed numerical results are presented in Section 3
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and finally, conclusions are drawn and suggestions for further research are given in Section 4. The Proposed Methodology
The VRPDDP aims to design a service network of vehicles for delivering goods from a depot to customers and picking up goods from customers. In this problem, each customer may demand only
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pickups, only deliveries or both. Since individual delivery or pickup demands of customers cannot be split into smaller loads, deliveries and pickups can be separately executed by visiting a customer
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twice (one for delivery and another for pickup) or simultaneously by visiting a customer once. All demands of each customer have to be satisfied by using identical capacitated vehicles. The mathematical model of the problem proposed by Nagy, Wassan [7] as a VRPMB by creating a pair of
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dummy co-located customers as deliveries and pickups for each customer. The MILP model of the problem is formulated by Nagy, Wassan [7] as in Appendix A.
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Exact approaches suggested for solving the VRP variants are not usually practical for large-sized problem instances because of the NP-hard nature of the problem [36]. Therefore, a cooperative
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variable neighborhood search (CVNS) algorithm is proposed for solving the VRPDDP. In this section, we first describe the general steps of the VNS algorithm, followed by its parallel variant proposed to solve the problem. 2.1. The variable neighborhood search Modern metaheuristic approaches commonly involve some kind of neighborhood structure for moving from one feasible solution to another. The VNS [29, 37, 38] is a powerful neighborhood structure based approach has been widely used in solving a variety of combinatorial optimization problems [39-41]. The main idea of the VNS approach is that a solution of one neighborhood
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ACCEPTED MANUSCRIPT structure does not have to be a local optimum of another neighborhood structure. The power of the VNS approach is built on the systematic exploration of a number of neighborhood structures used in shaking and local search steps, with the aim of reaching an improved solution. The shaking step increases the chances of finding a global optimum solution by a random selection strategy for a number of neighborhood structures (k-max) associated with the current best solution. Local search is an improvement procedure that makes use of a number of neighborhood structures (m-max). In this study, the variable neighborhood descent (VND) method, which explores neighborhoods one by one
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in a sequential order, is employed. The incumbent solution, which is obtained after the shaking and local search operators have been applied, is compared with the global solution in order to decide whether to move or not. Figure 3 shows the steps of the basic VNS used in this paper.
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The details of the proposed approach are presented in the following subsections as solution representation and objective function calculation, initial solution generation, stopping criteria definition, neighborhood structure design and parallelization strategy.
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2.2. Solution representation and calculation of the objective function
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In this study, an encoding mechanism, namely matrix representation, is used to represent VRP solutions proposed by the VNS (Figure 4). In this representation, the entire matrix represents a service network for the problem, each row in the matrix represents the route of a vehicle, each block
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in the row represents the node of a client and the cells in each block represent pickup and delivery operations. By using this encoding mechanism, efficient local search neighborhood structures can
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successfully be applied to the problem. The neighborhood structures in the shaking and local search phases first attempt to exchange clients with their blocks. If this attempt creates an infeasible
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solution, then block of a random client from this operation is divided into two new blocks. The pickup demand cell of this divided block assigned to one empty block and the delivery demand cell to another empty block. Thus, exchange operations are performed on these new blocks. Note that, a block may contain delivery or pickup demands or both of each client on its cells.
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ACCEPTED MANUSCRIPT Firstly, the total delivery and pickup loads are separately checked in the calculation of the fitness value. If both of the loads stay under the vehicle capacity, then the load of the vehicle at each client is further controlled step-by-step. During this control, if there is no capacity exceeding load in the departure of the vehicle from the client, than, the route called as a feasible route. The fitness value of the feasible route is calculated as the total travel distance. If any of these loads exceeds the capacity of vehicle, the route is penalized with a penalty value in the shaking step in order to give this route an additional chance in the local search. In this case, the violation limit (α) is used. If this route
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stays within the allowed violation limit, this route is given a very high penalty value. On the other hand, if the total load exceeds the violation limit (α), the route is categorized as an infeasible solution. However, violations are not permitted in the local search, and only feasible solutions are admitted, shared and compared with the current best solutions.
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2.3. Initial solution
The savings algorithm [42] is used to construct an initial solution. The algorithm aims to merge subtours based on costs savings achieved by combinations of two sub-tours. Some enhancements of the algorithm, depending on the type of problem, have provided an increase in the quality of the solution in the literature [43]. The extended savings formula (
In this equation,
,
di d j d
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Sij ci 0 c0 j cij c0i c j 0
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Equation (1) [44].
) between client i and j given in
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represent the distance between clients and the depot (0),
(1) and
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and represent the demand of clients and ̅ is the average demand. Here, λ, μ and ν are the route redesign, asymmetry of information and assignment priority parameters, respectively.
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2.4. Neighborhood structure
In this study, a number of intra and inter-route neighborhood structures commonly used in the
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literature [34, 45, 46] are employed in the shaking and local search steps in order to create a more flexible search within the solution space. In this content, swap, 2-opt, 3-opt and insertion are used as intra route neighborhood search operators and exchange (m,n), Cross, Shift (0,1) and Replace (1,1) structures are used as inter route search operators. A swap is a random position change movement of two customers on the same route. An insertion randomly selects a customer and moves it to a random position on the same route. The 2-opt operation swaps pairs of edges in the same route. The 3-opt deletes three edges in a route and reconnects them in the same route. An exchange (m,n) structure transfers m sequential customers
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ACCEPTED MANUSCRIPT from one route to another and in turn transfers n sequential customers from the second route to the first. Cross exchange is a basic crossover structure between routes. Shift (0,1) is a random movement of a customer from one route to another. Replace (1,1) is a random permutation movement between two customers from different routes. 2.5. Stopping condition The number of iterations or the total execution time of the algorithm are common stopping
of tours (shaking + local search) without improvement (s-max). 2.6. The parallel variable neighborhood search
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conditions of the VNS approach. In this research, the stopping condition implemented is the number
In the literature, various strategies have been executed and compared for parallelization of the VNS approach [25, 47, 48]. In this study, we used asynchronous cooperation with a centralized
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information exchange strategy in the parallelization of the VNS approach (see Figure 5). The details of the strategy is further explained by using the three-dimensional taxonomy of Crainic and Hail [22]; where the first dimension represents the search control cardinality, the second is the communication control and the third is the search differentiation. In this taxonomy, the developed approach is classified as pC/C/MPSS; where pC (p-Control) represents the global search, which is distributively
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controlled by several processes, C (Collegial) represents the data exchanged as asynchronously collegial and MPSS (Multiple initial Points, Same search Strategies) represents VNS approaches
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starting from different initial points and using the same neighborhood combinations and sequences.
In this strategy, the collegial data exchange is only performed after the completion of all local search procedures (medium-grained parallelization). Different initial solutions for each processor are
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constructed with savings heuristics by using different parameters. The acceptance criterion of the incumbent solution is to only accept improvements in the proposed VNS. This procedure, however, may cause the search to become stuck at a local optimum. Accepting unimproved solutions after a number of iterations (p-max), counting from the last accepted move, is a commonly used strategy for search diversification. In order to escape from a local optimum, an effective strategy, such as perturbation, is required in order to search promising regions. In this study, we used Double Replace perturbation operators, which are comprised of two sequential Replace (1,1) operations. Note that different initial solutions and perturbed solutions lead the search differentiation to the MPSS.
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ACCEPTED MANUSCRIPT In parallelization of the VNS, the star-architecture-based multiprocessor cooperation network, which uses a number of computation processors and a user interface processor, is employed. Figures 6 and 7 show the pseudocode for how this asynchronous cooperation with a centralized information exchange strategy is implemented. Figure 6 represents how each computation processor improves the solution. Figure 7 shows how the user interface processor communicates with other processors to update the best value and control the perturbation and stopping conditions without performing any computations. See Davidovid and Crainic *25+ for the details of asynchronous cooperation with a
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centralized information exchange strategy. In our adaptation, the initial solution, constructed with a savings heuristic, is refined by the basic VNS during a number of iterations, in addition to the perturbation mechanism for search diversification.
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Numerical Investigation
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In this section, we present the parameter settings of the proposed approach, explain the form of benchmark instances, provide the effect of parallelization of the approach on solution quality,
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compare the results of the approach with the best-known results from the literature and present the benefit of divisible delivery and pickup operations. The proposed CVNS was coded using MATLAB
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R2013b/Visual C++ 2010 and executed on a workstation computer with two Intel Xeon E5420 – 2.50 GHz Intel processors with four cores each and a total of 8 GB of RAM. The parallelization of the CVNS algorithm was performed using a limited number of processors, i.e., six cores. One core was used as a
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user interface processor and one for Windows and system operations.
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3.1. Parameter setting
The parameters in the savings algorithm extension of Altınel and Öncan *44+ are randomly drawn from the intervals λ ∈ [0, 5], μ ∈ [0, 3] and ν ∈ [0, 2] in order to construct different initial solutions for each processor. Note that the initial solutions in the VRPDDP are constructed as in the VRPSPD and improved with the basic VNS during N×10 non-improving loops, where N is the number of customers. The capacity violation limit (α) in the shaking step is set as 1.2, which gives the approach an additional chance to find feasible solutions in the local search phase. Global solutions are perturbed after N×1 non-improving loops (p-max), and the global search is stopped after N×500 non-improving loops (s-max) as a result of preliminary tests. Four (k-max) neighborhood structures - 3-opt, swap,
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ACCEPTED MANUSCRIPT exchange (m,n) and Cross - are applied in shaking, and four (m-max) neighborhood structures insertion, 2-opt, Shift (0,1) and Replace (1,1) - in local search in a given sequence. Whenever the proposed approach needs the perturbation mechanism, one of the mentioned perturbation operators is randomly applied to the current best solution. 3.2. Benchmark instances The benchmark instances used for the VRPSPD and the VRPDDP were generated from the VRP
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benchmark problems of Christofides, Mingozzi [49] by Salhi and Nagy [16] and Nagy, Wassan [7], respectively. Note that both data sets were reformulated differently in the study of Nagy, Wassan [7], contrary to the common usage in the literature. The authors introduced rounding to enable more correct comparisons, as rounding errors were making comparisons of results inconsistent. The authors use rounded values while generating delivery and pickup values from the original demands
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of Christofides, Mingozzi *49+ and while calculating the Euclidean distance between pairs of customers. Therefore, in this study the demand and the distance values are used in rounded form, as they are used by Nagy, Wassan [7] for comparison purposes. The original data set (CMT) of Salhi and Nagy [16] includes 14 test instances with 50–199 customers (X series) and an additional 14 instances (Y series) which were generated by exchanging pickup and delivery values for every other customer.
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While the CMT 1-2-3-4-5-11-12 X&Y instance group includes only capacity constraints for the vehicles, the CMT 6-7-8-9-10-13-14 X&Y instance group, which uses the same demand and location
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values as the first group, includes additional time-limit constraints for completing routes. In addition to the original CMT data set, Nagy, Wassan [7] proposed two new data sets using the same customer locations as the original one. If the demands in the original set are defined as fine-grained (Set 1), the
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demands in the new sets would be defined as medium- (Set 2) and coarse-grained (Set 3). Service time in the instances with a time limit is set at half of the original amount for each visit when the
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client is served twice in the VRPDDP. Note that both problem types can use the same benchmark instances with the same objective function, subject to different constraints. Addition to VRPDDP data
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instances generated by Nagy, Wassan [7], well-known VRPSPD benchmark data sets generated by Dethloff [50] and Tang Montané and Galvão [51] are solved by considering VRPDDP constraints first the time in the literature. Dethloff [50]’s benchmark set (SCA-CON) contains 40 instances with 50 customers and Tang Montané and Galvão [51]’s set (MON-GAL) contains 18 instances with 100–400 customers. However, we only considered 12 instances with up to 200 customers in MON-GAL set. 3.3. Effect of parallelization The performance of the proposed parallel VNS approach is compared with perturbation-based VNS, whose efficiency was shown by Polat, Kalayci [34]. The authors compared the performance of the
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ACCEPTED MANUSCRIPT PVNS on a number of VRPSPD benchmark instances with various well-known metaheuristic adaptations from the literature. Therefore, in this study, the performance of both VNS extensions is compared with the help of the VRPSDP (Set 1) instances formulated as in Nagy, Wassan [7]. Table 1 shows performance comparisons between existing sequential and proposed parallel VNS extensions by using same computer configuration.
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The results show that the parallel CVNS approach outperforms the PVNS in terms of both solution quality and CPU time. The CVNS approach provides better solutions for 10 of 28 test instances and the same results for the remaining instances within almost half the solution times of the PVNS. The
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overall improvement of the CVNS over the PVNS is just around 0.14%. This improvement is around 0.11% for the instances without a time limit and 0.16% for the time-limited instances. Indeed, the CVNS shows the lowest average solution time for the considered instances. Figure 7 depicts the improvement on the total distance provided by the CVNS and the PVNS approaches for the same 11X and 13X instances in detail. It is observed that the CVNS approach provides better results in terms of
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both solution and time for the 13X instance and provides the same result in less time for the 11X instance. From Table 1 and Figure 8, it can generally be concluded that the CVNS algorithm produces
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adequate solutions in acceptable solution times over the VRPSPD Set 1 benchmark instances.
3.4. Computational results
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The performance of the proposed CVNS approach is compared with the heuristic approaches found in the literature. Salhi and Nagy [16] proposed Cluster Insertion Heuristics (CIH), Tang Montané and Galvão [51] tabu search (TS) and Wassan, Wassan [20] reactive tabu search (RTS) method for solving the CMT sets with rounded demands and distances. The RTS methodology of Wassan, Wassan [20] was later adapted by Nagy, Wassan [7] to solve the VRPDDP type. Recently, Nagy, Wassan [7] provided eight different versions for improving the RTS methodology (IRTS) of Wassan, Wassan [20] for the VRPDDP type. These eight versions - DVA, DVO, DSA, DSO, NVA, NVO, NSA and NSO - are symbolized in our study as IRTS(#), where # represents the given sequence of the version. Also, Zachariadis, Tarantilis [52] proposed guided tabu search (GTS), Zachariadis, Tarantilis [53] variable
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ACCEPTED MANUSCRIPT length BoneRoute (VLBR), Subramanian, Drummond [54] parallel iterated local search with a random neighborhood ordering (P-ILS-RVND), Souza, Mine [55] iterated local search and GENIUS (GENILS) and Zachariadis and Kiranoudis [56] arc promise algorithm (APA) for solving a set of benchmark instances including SCA-CON and MON-GAL sets. First, the performance of the heuristics are presented for solving the VRPSPD type over three CMT sets of problem instances for comparison purposes. CMT Set 1 instances were executed by the CIH, TS and RTS approaches, and CMT Set 2 and 3 instances were executed by the RTS approach. Table 2
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shows the overall comparisons of the results provided by the CVNS approach and the best known results from the heuristic approaches defined above. The computational details of the results for CMT Sets 1, 2 and 3 are given in Tables B.1, B.2 and B.3 in the Appendix B, respectively.
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The results in the tables indicate that the CVNS provides the new best solutions for 26 out of 28 problem instances in CMT Set 1 and able to reach the best known solution in one instance. For the remaining instance, the gap between the best known solution and the proposed approach is only
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0.19%. The overall improvement of the proposed approach is just around 2.42%. This improvement is around 2.85% for the instances without a time limit and 1.98% for those with a time limit. The
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proposed approach provides the new best solutions for all instances of CMT Set 2, with a 2.40% improvement on the total distance and a 1.14 unit decrease on the total number of vehicles on
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average. The new best solutions are presented in 18 out of 28 instances and best known solutions are reached for the remaining CMT Set 3 instances. The average improvement on the total distance is
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around 0.88% and the average decrease on the vehicle number is around 1.71 units. Second, the performances of the heuristics are compared for solving the VRPDDP type over three
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sets of problem instances. CMT Set 1 and Set 2 instances were executed by the RTS approach and IRTS versions and Set 3 instances were executed by the RTS approach. Table 3 shows overall comparisons of the results provided by the CVNS approach and the best known results from the heuristic approaches defined above. The computational details of the results for CMT Sets 1, 2 and 3 are given in Tables B.4, B.5 and B.6 in the Appendix B, respectively.
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ACCEPTED MANUSCRIPT The results show that the CVNS provides the new best solutions for 27 out of 28 problem instances in CMT Set 1 and achieves the best known solution for the remaining instance. The improvement of the proposed approach in the total distance is around 3.67% and the decrease in the total vehicle number is around 0.36 units. The distance improvement is around 4.19% for instances without a time limit and 3.14% for those with a time limit. The proposed approach provides new best solutions for all of the instances in CMT Sets 2 and 3, providing 2.67% and 10.97% improvements on the total distance, respectively. While the decrease in the total vehicle number was 0.54 units in Set 2, it was
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11.57 units in Set 3.
The results also show that while the distance improvements decrease from fine-grained (CMT Set 1) to coarse-grained (CMT Set 3), the decrease of the number of total vehicles improves in the VRPSPD. On the other hand, the performance of the CVNS is very high for the VRPDDP, especially for coarse-
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grained instances.
Later, SCA-CON and MON-GAL sets are solved by both considering VRPSPD and VRPDDP types. The computational details of the results for SCA-CON and MON-GAL sets are given in Tables B.7 and B.8 in the Appendix B, respectively. In these tables, the results of the CVNS approach in the VRPSDP type are firstly compared with best-known results from the literature, then; the VRPDDP type results are
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compared with VRPSDP type best-known results from the literature. The average percentage gap of the solutions of CVNS from the optimal solutions [57] are 0.00% for SCA-CON set for VRPSPD type.
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The results also show that 9 out of 32 instances are improved 0.41% on average in VRPDDP compared to VRPSPD type. It was observed that improved SCA-CON instances have relatively greater number of vehicle usage compared to other instances. On the other hand, in MON-GAL set, the CVNS
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was able catch the optimal [57, 58] or the best-known solutions [54, 55] for 10 out of 12 instances. In VRPDDP type, the CVNS improved 3 out of 12 instances by 0.05% and achieved the same results for
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eight instances compared to the best-known results of VRPSPD type in the literature. Clustered customer locations are common property of improved MON-GAL instances in VRPDDP type.
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Table 4 summarizes the performance comparisons and solution CPU times of the CVNS approach with other heuristics from the literature for all sets. It is worth to note that it is not fair to compare solution CPU times of different algorithms since they were executed on different computer configurations, developed with different programming languages and coded with different programming skills. However, the processors of hardware configurations are scaled in Table 5 according to gigaFLOPS (GFLOPS) per core scores in order to give an idea regarding CPU performance of each approach compared in this study. Thus, solution CPU times in Table 4 is updated according to performance scores of related hardware configurations.
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The improvement values in CMT sets confirm that the CVNS approach outperformed all of the other heuristics for both VRPSPD and VRPDDP within acceptable solution times. Overall, when the average
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of solutions for the instances are compared, the CVNS provides 10.52% better solutions than CIH, 4.26% better than TS and 2.71% better than RTS on average for the VRPSPD. On the other hand, the CVNS provides 6.52% better solutions than RTS and 3.31% better solutions than the best solutions of eight versions of IRTS for the VRPDDP. For the SCA-CON set in VRPSPD type, the CVNS provides 0.75% better solutions than TS and 0.03% better than GTS and optimal solutions as in VLBR, P-ILS-RVND and
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GENILS. On the other hand, the CVNS provides 2.99% better solutions than TS, 0.55% better solutions than GTS, 0.20% better solutions than VLBR, 0.02% better solutions than GENILS and 0.15% better solutions than APA for the MON-GAL set in VRPSPD type. However, P-ILS-RVND is better in two instances of MON-GAL set by only 0.04% in which CVNS has relatively high solution time. The results in the Appendix also show that the CVNS provides robust solutions considering the gap between the
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average of the best solutions and the average of average results.
The solutions of the CVNS for both the VRPSPD and VRPDDP in all sets are also compared in terms of
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the total travel distance and the total number of vehicles used. Table 6 compares the results in the
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Table B.1-8 in the Appendix B, respectively.
The results indicate that allowing divisible delivery and pickup demands provides efficient solutions,
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as was theoretically proven by Nagy, Wassan [7]. As expected, the VRPDDP provided better solutions in a majority of the instances and the same solutions for a couple of instances with fine-grained demands in CMT instances. The VRPDDP found better solutions for 10 of 28 instances in CMT Set 1 and the same solutions as the VRPSPD for the rest. Despite the average distance being improved 0.07% on average, the decrease in the total number of vehicles is almost zero, except in the 14Y instance, which has less total distance with one more vehicle in the VRPDDP. For the CMT Set 2 and 3 instances, the total distance was improved by 2.67% and 10.97% by the VRPDDP, respectively. While the decrease in the number of vehicles is 1.71 units in CMT Set 2, it is 24.29 units for CMT Set 3. The
15
ACCEPTED MANUSCRIPT VRPDDP found better solutions for 9 of 40 instances in SCA-CON set and the same results as the VRPSPD for the rest. The CVNS results in VRPDDP type are 0.03% better than compared to the CVNS results in VRPSPD type for 4 of 12 instances of MON-GAL set. The results show that the performance of the VRPDDP highly depends on the size and differentiation of the demands for the problem instances. While the problem provides high benefits in terms of the total distance and vehicles for coarse- and medium-grained demands, the benefit is very limited for the instances with fine-grained
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demands.
Conclusion
In this paper, we proposed a parallel variable neighborhood search approach to solve the vehicle routing problem with divisible deliveries s and pickup (VRPDDP). In the parallelization of the variable
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neighborhood search approach, asynchronous cooperation with a centralized information exchange strategy was used. The main characteristics of the CVNS approach are the problem-specific designed encoding mechanism and neighborhood procedures and search strategy which allows parallelization along with perturbation. A number of benchmark instances known from vehicle routing problem instances were used to test the proposed approach. The obtained results were used to compare the
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performance of the proposed approach with well-known heuristics from the literature. The numerical investigations indicated that the proposed approach improved on or reached the best
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known solutions for 179 of 220 benchmark instances. The results also show that the CVNS provided these solutions while using less computational time and keeping its robustness. The proposed approach can be adapted to a variety of vehicle-routing problem applications. Future studies may
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also include memory-based perturbation strategies in order to increase the efficiency of the
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diversification of the approach. Acknowledgment
The author is grateful to area editor, anonymous reviewers and Dr. Gabor Nagy for their constructive
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comments and suggestions that helped improve the quality and exposition of this work. This research is funded by the Scientific Research Project Coordination Unit of Pamukkale University (PAUBAP) with the grant number 2014BSP012.
16
ACCEPTED MANUSCRIPT Appendix A: Mathematical model
Sets
D
The set of depots (consisting of a single depot):
L
The set of delivery customers:
0
1, 2,..., n
B
The set of pickup customers (pickup customer n+i, copy of delivery customer i):
V
The set of all locations: D L B
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n 1, n 2,..., 2n
Parameters: the maximum load capacity of a vehicle
d ij
the distance between location i and j (the distance between pickup customer n+i and delivery customer i is zero)
qi
demand of customer i
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C
Decision variables:
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1, if the edge between client i and j belongs to a route;
xij
0, otherwise.
Delivery loads transported between location i and j
Pij
Pickup loads transported between location i and j
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Rij
The model formulation is given as follows:
d x
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min
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s.t.
(2)
ij ij
iV jV
x iV
ij
x iV
ji
R iV
ij
1 j L B
(3)
1 j L B
(4)
q j R ji
(5)
j L
iV
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R R ij
j B
ji
Ri 0 0
(7)
P q P ij
iV
j
ij
iL B
iV
ji
j B
(8)
j L
(9)
P0i 0
Rij Pij Cxij C
ji
iV
P P iV
(6)
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iL B
iV
iL B
i, j V
x0i qi iL
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iV
xii 0 i V
x( ni )i 0 i L
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xii 0,1 , Rij 0, Pij 0 i, j V
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xij x ji 1 i, j L B
(10) (11) (12) (13) (14) (15) (16)
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The objective function (2) aims to minimize the total travel distance. Equations (3) and (4) ensure that each customer is served once. Constraints (5)-(7) are flow conservation constraints for pickups. Constraints (8)-(10) are flow conservation constraints for deliveries. Constraint (11) is the vehicle capacity constraint. Restriction (12) guarantees usage of the minimum number of vehicles. Constraints (13)-(15) are additional preclusive loop constraints. Finally, constraint (16) defines the variable domains.
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Appendix B: Detailed results
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References
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ACCEPTED MANUSCRIPT [44] Altınel K, Öncan T. A new enhancement of the Clarke and Wright savings heuristic for the capacitated vehicle routing problem. Journal of the Operational Research Society. 2005;56(8):954–61 [45] Li K, Tian H. A two-level self-adaptive variable neighborhood search algorithm for the prizecollecting vehicle routing problem. Applied Soft Computing. 2016;43:469-79. [46] Khouadjia MR, Sarasola B, Alba E, Jourdan L, Talbi E-G. A comparative study between dynamic adapted PSO and VNS for the vehicle routing problem with dynamic requests. Applied Soft Computing. 2012;12(4):1426-39.
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*47+ Crainic TG, Gendreau M, Hansen P, Mladenovid N. Cooperative Parallel Variable Neighborhood Search for the p-Median. Journal of Heuristics. 2004;10(3):293-314. [48] García-López F, Melián-Batista B, Moreno-Pérez J, Moreno-Vega JM. The Parallel Variable Neighborhood Search for the p-Median Problem. Journal of Heuristics. 2002;8(3):375-88.
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*49+ Christofides N, Mingozzi A, Toth P. The vehicle routing problem. In: Christofides AM, P. Toth, & C. Sandi (Ed.). Combinatorial optimization. Chichester: Wiley; 1979. p. 315-38. [50] Dethloff J. Vehicle routing and reverse logistics: the vehicle routing problem with simultaneous delivery and pick-up. OR Spektrum. 2001;23:79-96. [51] Tang Montané FA, Galvão RD. A tabu search algorithm for the vehicle routing problem with simultaneous pick-up and delivery service. Computer and Operations Research. 2006;33:595-619.
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[52] Zachariadis EE, Tarantilis CD, Kiranoudis CT. A hybrid metaheuristic algorithm for the vehicle routing problem with simultaneous delivery and pick-up service. Expert Systems with Applications. 2009;36( 2/1):1070-81.
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[54] Subramanian A, Drummond LMA, Bentes C, Ochi LS, Farias R. A parallel heuristic for the vehicle routing problem with simultaneous pickup and delivery. Computers and Operations Research. 2010;37:1899-911.
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[55] Souza MJF, Mine MT, Silva MdSA, Ochi LS, Subramanian A. A hybrid heuristic, based on Iterated Local Search and GENIUS, for the Vehicle Routing Problem with Simultaneous Pickup and Delivery. International Journal of Logistics Systems and Management. 2011;10(2):142-57. [56] Zachariadis EE, Kiranoudis CT. A local search metaheuristic algorithm for the vehiclerouting problem with simultaneous pick-ups and deliveries. Expert Systems with Applications. 2011;38(3):2717-26. [57] Subramanian A, Uchoa E, Pessoa AA, Ochi LS. Branch-and-cut with lazy separation for the vehicle routing problem with simultaneous pickup and delivery. Operations Research Letters. 2011;39(5):338-41.
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[58] Subramanian A, Uchoa E, Pessoa AA, Ochi LS. Branch-cut-and-price for the vehicle routing problem with simultaneous pickup and delivery. Optimization Letters. 2013;7(7):1569-81.
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Figure 1: Basic subclasses of VRPPD: (a) VRPCB, (b) VRPMB, (c) VRPSPD, (d) VRPDDP
23
ACCEPTED MANUSCRIPT Route
O →A →B →C →D →E →F →G →O
Route
Delivery
16
8
10
8
7
5
20
Pickup
15
3
20
20
18
15
5
Effect
-1
-5
10
12
11
10
-15
Effect
73
68
78
90 101 111
96
Load
5
10
15
20
35
Load
74
Distance
25
30
O →G →F →E →D →C →B →A →O
Delivery Pickup
45
74
Distance
20
5
7
8
10
8
16
5
10
18
20
20
3
15
-15
10
11
12
10
-5
-1
59
69
80
92 102
97
96
10
15
20
25
35
40
O →A →B →C →D →G →E
→F →O
Delivery
16
8
10
8
20
7
5
Pickup
15
3
20
20
5
18
Effect
-1
-5
10
12
-15
73
68
78
90
5
10
15
20
Load
74
Distance
Route
O →A →B →C →D →E.d →F →G →E.p →O
Delivery
16
8
10
8
7
5
20
0
15
Pickup
15
3
20
20
0
15
5
18
11
10
Effect
-1
-5
10
12
-7
10
-15
18
75
86
96
Load
73
68
78
90
83
93
78
96
33
41
46
5
10
15
20
25
30
35
43
54
Distance
(d)
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(c)
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Route
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Figure 2: Alternative routes for VRPSPD and VRPDDP
Initialization: Construct an initial solution x
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Set k←1
45
(b)
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(a)
30
Loop: repeat following steps until stopping condition is satisfied Shaking: Select a solution x’ from kth neighborhood structure of solution x, Nk (k=1,2,…k-max) Local search: Apply the local search with solution x’ as initial solution and obtain incumbent solution x’’ Move or not: If x’’ is better than x, set x← x’’ and k←1; otherwise, set k←k+1
Figure 3: The phases of the basic VNS
24
48
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Figure 4: Encoding of solutions
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Figure 5: Basis of asynchronous cooperation with centralized information exchange strategy
25
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Initialization: Execute the savings heuristics to obtain an initial solution x Refine: Improve x by using basic VNS to obtain incumbent solution x’’ Communication 1: Share x’’ with user interface and receive xbest Set k←1 Loop: repeat following steps until receiving STOP message
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Shaking: Select a solution x’ from kth neighborhood structure of solution x, Nk (k=1,2,…k-max)
Local search: Apply the local search with solution x’ as initial solution and obtain incumbent solution x’’ Communication 2: Share x’’ with user interface and check the message from user interface If STOP message is arrived exit the loop; otherwise, receive xbest from user interface and update k
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Figure 6: The pseudo code of improvement phases for each computation processor in CVNS
Communication 1: Receive x’’ from all processors and share best of them as xbest with all processors Loop: repeat following steps until providing STOP message
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Communication 2: Receive x’’ from all processors If x’’ is better than xbest , set xbest ← x’’, share xbest with all processors and reset p-max and s-max counters to 1; otherwise, update p-max and s-max counters. Perturbation: If perturbation condition is met, share perturbed xbest with all processors
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Stopping: If stopping condition is met, send STOP message to all processors
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Figure 7: The pseudo code of communication phases for user interface processor in CVNS
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Table 1: Performance compression between PVNS and CVNS for VRPSPD (CMT Set 1) PVNS
CVNS
k
Best sol.
Avg. sol.
Avg. t
Best sol.
Avg. sol.
Avg. t
Imp(%)
1X
3
470
470.0
19.9
470
470.0
9.7
0.00
1Y
3
459
459.0
10.0
459
459.0
5.7
0.00
2X
6
685
685.0
54.8
685
685.0
24.4
0.00
2Y
6
651
651.0
57.9
651
651.0
30.3
0.00
3X
5
715
715.4
62.7
714
714.4
70.3
0.14
3Y
5
709
709.0
57.1
705
705.6
46.1
0.56
4X
7
866
866.3
145.7
862
863.1
156.0
0.46
4Y
7
831
831.0
167.6
831
831.0
92.3
0.00
5X
10
1063
1063.0
689.1
1063
1063.0
310.0
0.00
5Y
9
985
985.8
357.4
982
983.3
280.7
0.30
6X
6
548
548.0
57.5
548
548.0
29.2
0.00
6Y
6
548
548.0
56.8
548
548.0
23.2
0.00
7X
11
897
897.0
87.4
897
897.0
49.1
0.00
7Y
11
897
897.0
84.2
897
897.0
45.5
0.00
8X
9
8Y
9
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Instance
863.0
270.0
856
858.1
229.4
0.81
863
863.0
197.8
856
856.6
191.7
0.81
14
1143
1143.8
587.3
1143
1143.8
238.7
0.00
14
1143
1143.4
581.6
1143
1143.4
271.7
0.00
10X
18
1374
1375.3
1461.0
1373
1373.7
1181.0
0.07
10Y
18
1369
1370.0
1350.1
1366
1366.9
1568.1
0.22
11X
4
874
874.0
40.8
874
874.0
20.6
0.00
11Y
4
826
826.8
61.2
826
826.8
33.8
0.00
12X
6
672
672.1
41.8
672
672.1
24.3
0.00
12Y
5
632
633.9
40.2
632
633.9
22.1
0.00
9X
AC
9Y
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863
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11
1555
1555.0
410.9
1551
1552.3
335.4
0.26
13Y
11
1550
1550.5
463.9
1549
1549.5
518.4
0.06
14X
9
821
821.0
282.9
821
821.0
151.4
0.00
14Y
9
821
821.0
250.8
821
821.0
120.2
0.00
886.79
887.08
283.9
885.54
886.02
217.1
0.14
Average
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k: Number of vehicles; Best sol.: Best solution; Avg. sol.: Average solution; Avg. t.: Average best solution finding CPU time in seconds; Imp(%): Percentage improvement between the PVNS and the CVNS which is calculated as ((Best sol. (PVNS) - Best sol. (CVNS)) / Best sol. (PVNS)) × 100; Average: Average of 28 instances; Bold values indicate that CVNS found the best known or the new best solutions.
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ACCEPTED MANUSCRIPT Table 2: Overview of computational results for VRPSPD (CMT) CMT Set 1 Δk
Imp(%)
Δk
Imp(%)
Δk
1X
0.42
0
1.01
0
0.00
0
1Y
2.34
0
1.45
0
1.87
1
2X
1.44
0
2.36
2
0.00
0
2Y
6.20
0
3.99
2
0.00
0
3X
0.97
0
1.90
1
1.40
3
3Y
1.95
0
2.18
0
0.96
3
4X
2.05
0
1.25
1
1.26
3
4Y
3.26
0
1.26
0
1.05
1
5X
2.48
0
5Y
6.74
1
6X
1.26
0
6Y
1.26
-1
7X
0.22
0
7Y
0.55
0
8X
1.95
8Y
1.27
AN US
CR IP T
Imp(%)
3
1.46
5
1.67
1
0.52
2
1.01
0
0.00
0
1.45
0
1.87
1
2.78
2
0.00
0
3.99
2
0.00
0
0
1.90
1
1.40
3
0
2.18
0
0.96
3
3.79
0
1.25
1
1.26
3
3.79
0
1.26
0
1.05
1
10X
3.31
0
1.91
3
1.46
5
10Y
3.80
-1
1.67
1
0.52
2
11X
2.89
0
4.39
2
2.79
3
11Y
8.73
0
4.59
1
1.05
3
12X
0.44
-1
2.53
2
0.00
0
12Y
0.00
0
2.88
1
0.00
0
13X
0.39
0
4.39
2
2.79
3
13Y
-0.19
0
4.59
1
1.05
3
PT
CE
9Y
ED
1.91
9X
AC
CMT Set 3
M
Instance
CMT Set 2
29
ACCEPTED MANUSCRIPT 14X
2.49
1
2.53
2
0.00
0
14Y
3.86
1
2.88
1
0.00
0
Average
2.42
0.00
2.40
1.14
0.88
1.71
AC
CE
PT
ED
M
AN US
CR IP T
Imp(%): Percentage improvement between BKS and the CVNS which is calculated as ((BKS - Best Sol. (CVNS)) / BKS) × 100; Δk: Deviation in the number of total vehicle number in the network; Bold values indicate that CVNS found the best known or the new best solutions.
30
ACCEPTED MANUSCRIPT Table 3: Overview of computational results for VRPDDP (CMT) CMT Set 1
Imp(%)
Δk
Imp(%)
Δk
1X
1.67
0
1.90
1
5.97
3
1Y
3.57
0
1.67
0
6.45
4
2X
3.93
1
1.31
2
14.77
12
2Y
6.34
1
3.50
2
14.67
15
3X
1.79
0
2.21
0
10.33
10
3Y
2.49
0
1.63
0
8.13
6
4X
4.33
1
1.40
0
14.03
18
4Y
3.26
0
1.39
0
7.15
10
5X
1.67
1
5Y
6.65
1
6X
1.26
0
6Y
1.44
0
7X
0.22
0
7Y
0.11
0
8X
2.06
8Y
1.27
AN US
CR IP T
Δk
1
14.63
28
1.67
1
11.14
24
1.90
1
5.97
3
1.67
0
6.45
4
1.91
1
14.77
12
3.50
2
14.67
15
0
2.21
0
10.33
10
0
1.63
0
8.13
6
ED
M
0.19
9X
2.48
0
1.40
0
14.03
18
9Y
5.93
1
1.39
0
7.15
10
10X
3.99
0
0.19
1
14.63
28
10Y
5.73
1
1.67
1
11.14
24
11X
13.48
1
5.78
0
13.72
9
11Y
8.73
0
3.96
0
10.43
7
12X
0.74
0
1.98
0
12.67
7
12Y
0.00
0
2.96
1
9.46
9
13X
5.96
0
5.78
0
13.72
9
13Y
9.19
0
3.96
0
10.43
7
CE AC
CMT Set 3
Imp(%)
PT
Instance
CMT Set 2
31
ACCEPTED MANUSCRIPT 0.24
1
1.98
0
12.67
7
14Y
4.10
1
2.96
1
9.46
9
Average
3.67
0.36
2.28
0.54
10.97
11.57
AC
CE
PT
ED
M
AN US
CR IP T
14X
32
ACCEPTED MANUSCRIPT Table 4: Performance improvement of the CVNS compared to other heuristics CIH
TS
RTS
GTS
VLBR
Imp (%)
10.52
4.26
4.51
-
-
t
2.39
st
0.08
5.13
20.71
Imp (%)
-
-
2.55 94.03
st
17.55 -
-
t
MONGAL
3.73
st
1.00
st Imp (%)
-
-
-
2.99 34.17
-
-
9.14 -
PT
-
4.41
-
-
-
-
2454.79
-
-
-
-
320.60 320.60
5584.14 81.08 81.08
0.03
0.00
0.00
0.00
-
-
758.54
3.44
3.27
2.75
8.83
6.16
0.71
1.14
2.06
3.08
6.16
0.55
0.20
-0.04
0.02
0.15
- 1712.56
33.55
26.88
48.88
37.48
87.37
264.29
6.97
9.37
36.66
13.07
30.46
264.29
-
-
-
-
-
4.18
884.79
121.57
406.80
st
23.79
22.69
406.80
CE
AC VRPDDP
885.54
127.46
-
-
3.61
-
-
-
-
-
2.44 2384.43
t
107.00
100.61
814.70
st
19.97
18.78
814.70
-
4409.00
Imp (%)
CMT Set 3
-
t
Imp (%)
CMT Set 2
-
t
CMT Set 1
0.75
t
Imp (%)
-
M
SCACON
-
-
217.10
-
st Imp (%)
CVNS
217.10
1.05
ED
VRPSPD
CMT Set 3
Best in IRTS
19.17 110.96
t
Imp (%)
-
APA
CR IP T
CMT Set 2
-
AN US
CMT Set 1
P-ILSRVND GEN ILS
-
-
t
11.53
-
-
-
-
-
-
928.11
st SCA- Imp (%) CON t
928.11 -
-
-
-
-
-
-
-
-
757.78 23.96
33
ACCEPTED MANUSCRIPT st Imp (%) MONGAL
23.96 -
-
-
-
-
-
-
-
- 1711.04
t
990.01
st
990.01
AC
CE
PT
ED
M
AN US
CR IP T
Imp(%): Percentage improvement between the related heuristic and CVNS which is calculated as ((Best sol. heuristic – Best sol. (CVNS)) / Best sol. heuristic)×100, t: original average solution time reported in related article; st: scaled average solution time according to performance scores in Table 5 (st=t×Score)
34
ACCEPTED MANUSCRIPT Table 5: CPU performance comparison. Algorithm
Author
CPU used to implement
Programming Language
GFLOPS/
a
Score core
CIH
Salhi and Nagy [16]
VAX 4000-500 71.4 MHz
Fortran 77
0.09
TS
Tang Montané and Galvão [51]
Athlon XP 2.0 GHz
Delphi 5.0
0.76
RTS
Wassan, Wassan Sun-Fire-V440 [20] UltraSPARC-IIIi 1.06 GHz
Fortran 90
GTS
Zachariadis, Tarantilis [52]
Pentium IV 2.4 GHz
Visual C#
VLBR
Zachariadis, Tarantilis [53]
Intel Core 2 Duo 1.66 GHz
Visual C#
0.99
P-ILS-RVND
Subramanian, Drummond [54]
Intel Xeon 2.66 GHz
C++
2.13
GENILS
Souza, Mine [55] Intel Core 2 Duo 1.66 GHz
Visual C++ 2005
0.99
APA
Zachariadis and Kiranoudis [56]
Visual C#
0.99
IRTS
Nagy, Wassan [7]
Sun-Fire-V440
ED
0.03
d
0.27
CR IP T
AN US
M
Intel Core 2 Duo 1.66 GHz
bc
Fortran 90
def
0.19
d
0.21
d
0.35
g
0.75
d
0.35
d
0.35
de
0.19
g
1.00
0.53
0.59
0.53
PT
UltraSPARC-IIIi 1.06 GHz
CVNS
This study
Intel Xeon E5420 2.50 GHz
Visual C++ 2010
Scaled CPU performance (GFLOPS of the compared CPU / GFLOPS of the CPU in this study)
b
Equivalent VAX 6000-520 62.5 MHz CPU is used
c
http://www.roylongbottom.org.uk/mips.htm#anchorDEC (Last accessed 15.12.2016)
d
http://www.roylongbottom.org.uk/linpack%20results.htm (Last accessed 15.12.2016)
e
Equivalent Pentium 4 1.9 GHz CPU is used
f
Recoded by Nagy, Wassan [7]
g
https://milkyway.cs.rpi.edu/milkyway/cpu_list.php (Last accessed 15.12.2016)
AC
CE
a
35
2.84
ACCEPTED MANUSCRIPT Table 6: Comparison of VRPSDP and VRPDDP results CMT Set 1
Inst. Imp (%)
CMT Set 2
CMT Set 3
SCA-CON
Δk
Imp (%)
Δk
Imp (%)
Δk
Inst.
1
MON-GAL
Imp (%)
Δk
Inst.
Imp (%)
Δk
0.00
0
2.98
1
21.08
13
SCA8-0
0.00
0
r101
0.00
0
1Y
0.00
0
2.49
1
16.43
13
SCA8-1
0.00
0
r201
0.00
0
2X
0.15
0
3.60
2
23.54
26
SCA8-2
0.00
0
c101
0.00
0
2Y
0.15
0
6.52
3
21.67
26
SCA8-3
3X
0.14
0
3.47
2
17.80
18
SCA8-4
3Y
0.00
0
2.06
1
16.59
19
SCA8-5
4X
0.00
0
3.72
2
20.23
31
SCA8-6
4Y
0.00
0
1.20
1
16.91
5X
0.09
0
3.14
4
21.77
5Y
0.00
0
1.46
2
18.29
6X
0.00
0
2.98
1
21.08
6Y
0.00
0
2.49
1
16.43
7X
0.00
0
3.60
2
23.54
7Y
0.00
0
6.52
3
8X
0.00
0
3.47
8Y
0.00
0
9X
0.09
0
9Y
0.00
0
10X
0.07
0
c201
0.00
0
0.00
0
rc101
0.04
0
0.00
0
rc201
0.00
0
0.24
0
r121
0.15
0
0.00
0
r221
0.00
0
43
SCA8-8
0.00
0
c121
0.19
0
43
SCA8-9
0.31
0
c221
0.00
0
13
CON8-0
0.01
0
rc121
0.03
0
13
CON8-1
0.19
0
rc221
0.00
0
26
CON8-2
0.89
0
21.67
26
CON8-3
0.00
0
2
17.80
18
CON8-4
0.12
0
2.06
1
16.59
19
CON8-5
0.00
0
3.72
2
20.23
31
CON8-6
0.00
0
1.20
1
16.91
28
CON8-7
0.00
0
0
3.14
4
21.77
43
CON8-8
0.07
0
CON8-9
1.40
0
ED
M
SCA8-7
PT
AN US
0.43
28
CE
AC
CR IP T
1X
10Y
0.00
0
1.46
2
18.29
43
11X
0.11
0
3.58
1
23.78
19
11Y
0.00
0
0.06
0
22.09
18
12X
0.00
0
4.60
3
25.86
22
12Y
0.00
0
2.52
1
23.80
21
13X
0.32
0
3.58
1
23.78
19
36
ACCEPTED MANUSCRIPT 13Y
0.45
0
0.06
0
22.09
18
14X
0.00
0
4.60
3
25.86
22
14Y
0.24
-1
2.52
1
23.80
21
Avg.
0.07
-0.04
2.96
1.71
20.70
24.29
0.18
0.00
0.03
1
0.00
AC
CE
PT
ED
M
AN US
CR IP T
Due to space constraints, only SCA8-* and CON8-* instances are shown in the table. Improvements for all of the remaining instances are 0.
37
ACCEPTED MANUSCRIPT Appendix B
Table B.1: Computational results for VRPSPD (CMT Set 1) Best known
CVNS
k
BKS
Ref.
k
Best sol.
Avg. sol.
Avg. t
Imp(%)
1X
3
472
TS
3
470
470.0
9.7
0.42
1Y
3
470
TS
3
459
459.0
5.7
2.34
2X
6
695
TS
6
685
685.0
24.4
1.44
2Y
6
694
RTS
6
651
651.0
30.3
6.20
3X
5
721
RTS
5
714
714.4
70.3
0.97
3Y
5
719
RTS
5
705
705.6
46.1
1.95
4X
7
880
RTS
7
862
863.1
156.0
2.05
4Y
7
859
RTS
5X
10
1090
RTS
5Y
10
1053
RTS
6X
6
555
CIH, RTS
6Y
5
555
7X
11
899
7Y
11
902
8X
9
873
8Y
9
9X
14
AN US
CR IP T
Instance
831
831.0
92.3
3.26
10
1063
1063.0
310.0
2.48
9
982
983.3
280.7
6.74
6
548
548.0
29.2
1.26
CIH
6
548
548.0
23.2
1.26
RTS
11
897
897.0
49.1
0.22
RTS
11
897
897.0
45.5
0.55
CIH
9
856
858.1
229.4
1.95
867
CIH, RTS
9
856
856.6
191.7
1.27
1188
CIH
14
1143
1143.8
238.7
3.79
CE
PT
ED
M
7
14
1188
CIH
14
1143
1143.4
271.7
3.79
10X
18
1420
CIH
18
1373
1373.7
1181.0
3.31
10Y
17
1420
CIH
18
1366
1366.9
1568.1
3.80
11X
4
900
TS
4
874
874.0
20.6
2.89
11Y
4
905
RTS
4
826
826.8
33.8
8.73
12X
5
675
TS
6
672
672.1
24.3
0.44
12Y
5
632
RTS
5
632
633.9
22.1
0.00
13X
11
1557
CIH
11
1551
1552.3
335.4
0.39
AC
9Y
38
ACCEPTED MANUSCRIPT 13Y
11
1546
CIH
11
1549
1549.5
518.4
-0.19
14X
10
842
RTS
9
821
821.0
151.4
2.49
14Y
10
854
RTS
9
821
821.0
120.2
3.86
885.54
886.02
217.1
2.42
Average
908.25
AC
CE
PT
ED
M
AN US
CR IP T
k: Number of vehicles; Best sol.: Best solution; Avg. sol.: Average solution; Avg. t.: Average best solution finding CPU time in seconds; Imp(%): Percentage improvement between the BKS and the CVNS which is calculated as ((BKS - Best sol. (CVNS)) / BKS) × 100; Average: Average of 28 instances; Bold values indicate that CVNS found the best known or the new best solutions.
39
ACCEPTED MANUSCRIPT
Table B.2: Computational results for VRPSPD (CMT Set 2) Best known
CVNS
k
BKS
Ref.
k
Best sol.
Avg. sol.
Avg. t
Imp(%)
1X
19
1185
RTS
19
1173
1173.0
16.4
1.01
1Y
18
1101
RTS
18
1085
1085.0
11.2
1.45
2X
38
2074
RTS
36
2025
2025.5
45.4
2.36
2Y
35
2028
RTS
33
1947
1947.0
32.1
3.99
3X
32
2055
RTS
31
2016
2018.0
246.0
1.90
3Y
28
1884
RTS
28
1843
1843.4
311.4
2.18
4X
47
2884
RTS
46
2848
2850.3
772.7
1.25
4Y
42
2612
RTS
5X
68
3865
RTS
5Y
59
3403
RTS
6X
19
1185
RTS
6Y
18
1101
7X
38
2083
7Y
35
2028
8X
32
2055
8Y
28
9X
47
AN US
CR IP T
Instance
2579
2582.6
418.4
1.26
65
3791
3791.0
737.0
1.91
58
3346
3352.4
508.8
1.67
19
1173
1173.0
16.4
1.01
RTS
18
1085
1085.0
11.2
1.45
RTS
36
2025
2025.5
45.4
2.78
RTS
33
1947
1947.0
32.1
3.99
RTS
31
2016
2018.0
246.0
1.90
1884
RTS
28
1843
1843.4
311.4
2.18
2884
RTS
46
2848
2850.3
772.7
1.25
42
2612
RTS
42
2579
2582.6
418.4
1.26
68
3865
RTS
65
3791
3791.0
737.0
1.91
10Y
59
3403
RTS
58
3346
3352.4
508.8
1.67
11X
32
3941
RTS
30
3768
3774.8
592.9
4.39
11Y
29
3333
RTS
28
3180
3180.6
353.7
4.59
12X
37
2609
RTS
35
2543
2544.5
259.3
2.53
12Y
33
2289
RTS
32
2223
2225.2
183.5
2.88
13X
32
3941
RTS
30
3768
3774.8
592.9
4.39
AC
10X
ED
PT
CE
9Y
M
42
40
ACCEPTED MANUSCRIPT 13Y
29
3333
RTS
28
3180
3180.6
353.7
4.59
14X
37
2609
RTS
35
2543
2544.5
259.3
2.53
14Y
33
2289
RTS
32
2223
2225.2
183.5
2.88
2454.79
2456.66
320.6
2.40
2519.11
AC
CE
PT
ED
M
AN US
CR IP T
Average
41
ACCEPTED MANUSCRIPT Table B.3: Computational results for VRPSPD (CMT Set 3) Best known
CVNS
k
BKS
Ref.
k
Best sol.
Avg. sol.
Avg. t
Imp (%)
1X
45
2315
RTS
45
2315
2315.0
20.7
0.00
1Y
45
2245
RTS
44
2203
2203.0
15.2
1.87
2X
74
3586
RTS
74
3586
3586.0
49.7
0.00
2Y
74
3586
RTS
74
3586
3586.0
39.2
0.00
3X
80
4210
RTS
77
4151
4154.7
88.2
1.40
3Y
80
4146
RTS
77
4106
4109.3
41.9
0.96
4X
121
6263
RTS
118
6184
6187.7
177.5
1.26
4Y
116
6017
RTS
115
5954
5959.4
112.6
1.05
5X
167
8366
RTS
5Y
163
8080
RTS
6X
45
2315
RTS
6Y
45
2245
RTS
7X
74
3586
7Y
74
3586
8X
80
4210
8Y
80
4146
9X
121
9Y
116
10X
AN US
CR IP T
Instance
8244
8246.5
54.6
1.46
161
8038
8040.4
43.0
0.52
45
2315
2315.0
20.7
0.00
44
2203
2203.0
15.2
1.87
RTS
74
3586
3586.0
49.7
0.00
RTS
74
3586
3586.0
39.2
0.00
RTS
77
4151
4154.7
88.2
1.40
RTS
77
4106
4109.3
41.9
0.96
6263
RTS
118
6184
6187.7
177.5
1.26
6017
RTS
115
5954
5959.4
112.6
1.05
167
8366
RTS
162
8244
8246.5
54.6
1.46
10Y
163
8080
RTS
161
8038
8040.4
43.0
0.52
11X
90
10024
RTS
87
9744
9744.0
161.5
2.79
11Y
89
9727
RTS
86
9625
9629.8
129.5
1.05
12X
83
5328
RTS
83
5328
5328.0
109.1
0.00
12Y
80
5114
RTS
80
5114
5114.0
92.5
0.00
13X
90
10024
RTS
87
9744
9744.0
161.5
2.79
13Y
89
9727
RTS
86
9625
9629.8
129.5
1.05
ED
PT
CE
AC
M
162
42
ACCEPTED MANUSCRIPT 14X
83
5328
RTS
83
5328
5328.0
109.1
0.00
14Y
80
5114
RTS
80
5114
5114.0
92.5
0.00
5584.14
5585.98
81.08
0.88
5643.4
AC
CE
PT
ED
M
AN US
CR IP T
Average
43
ACCEPTED MANUSCRIPT Table B.4: Computational results for VRPDPD (CMT Set 1) CVNS
k
BKS
Ref.
k
Best sol.
Avg. sol.
Avg. t
Imp (%)
1X
3
478
RTS, IRTS(1,2,3,5,6,7,8)
3
470
470.0
20.6
1.67
1Y
3
476
RTS, IRTS(1,2,3,4,5,6,7,8)
3
459
459.0
18.0
3.57
2X
7
712
RTS, IRTS(1,2,3,5,6,7,8)
6
684
684.8
204.2
3.93
2Y
7
694
RTS, IRTS(1,2,3,4,5,6,7,8)
6
650
651.0
256.8
6.34
3X
5
726
RTS, IRTS(1,2,3,4,5)
5
713
714.4
114.8
1.79
3Y
5
723
RTS, IRTS(1,2,3,4,5,6,7)
5
705
706.0
113.3
2.49
4X
8
901
RTS, IRTS(1,2,3,4,5,6,7,8)
7
862
862.0
40.2
4.33
4Y
7
859
RTS, IRTS(1,2,3,5,6,7)
7
831
831.0
80.3
3.26
5X
11
1080
IRTS(1)
10
1062
1063.6
287.9
1.67
5Y
10
1052
RTS, IRTS(3,7)
9
982
982.3
247.9
6.65
6X
6
555
RTS, IRTS(1,2,3,4,5,6,7,8)
6
548
548.0
58.5
1.26
6Y
6
556
RTS, IRTS(1,2,3,4,5,6,7,8)
6
548
548.0
85.4
1.44
7X
11
899
RTS, IRTS(1,2,5,6,7,8)
11
897
897.0
426.2
0.22
7Y
11
898
IRTS(1,3,5)
11
897
897.0
596.5
0.11
8X
9
874
RTS, IRTS(1,2,5,6)
9
856
857.8
507.7
2.06
8Y
9
867
RTS, IRTS(1,2,3,4,5,6,7,8)
9
856
857.5
433.6
1.27
9X
14
9Y
15
IRTS(1,5)
14
1142
1142.7
620.6
2.48
1215
RTS, IRTS(1,2,5,6,7)
14
1143
1144.4
761.5
5.93
18
1429
IRTS(1,3,5,7)
18
1372
1373.5
881.0
3.99
19
1449
IRTS(1,2,3,4)
18
1366
1367.5
948.7
5.73
11X
5
1009
RTS, IRTS(1,2,3,5,6,7)
4
873
873.9
476.6
13.48
11Y
4
905
RTS, IRTS(1,2,3,4,5,6,7,8)
4
826
827.3
529.3
8.73
12X
6
677
IRTS(7)
6
672
672.1
537.4
0.74
12Y
5
632
RTS, IRTS(1,2,3,4,5,6,7)
5
632
633.6
609.5
0.00
13X
11
1644
RTS, IRTS(1,2,5,6)
11
1546
1547.4
769.2
5.96
13Y
11
1698
IRTS(5,6,7)
11
1542
1542.8
639.4
9.19
10X 10Y
CE
1171
AC
PT
ED
AN US
CR IP T
Instance
M
Best known
44
ACCEPTED MANUSCRIPT 14X
10
823
IRTS(3,7)
9
821
822.6
548.8
0.24
14Y
11
854
RTS, IRTS(1,2,3,4,5,6)
10
819
819.4
577.2
4.10
884.79
885.59
406.8
3.67
923.43
AC
CE
PT
ED
M
AN US
CR IP T
Average
45
ACCEPTED MANUSCRIPT Table B.5: Computational results for VRPDPD (CMT Set 2) Best known
CVNS
k
BKS
Ref.
k
Best sol.
Avg. sol.
Avg. t
Imp(%)
1X
19
1160
IRTS(1)
18
1138
1139.8
71.5
1.90
1Y
17
1076
IRTS(5)
17
1058
1058.2
35.9
1.67
2X
36
1978
RTS
34
1952
1953.6
408.4
1.31
2Y
32
1886
IRTS(1,3)
30
1820
1820.9
513.6
3.50
3X
29
1990
IRTS(6,8)
29
1946
1946.6
229.5
2.21
3Y
27
1835
IRTS(3)
27
1805
1806.6
226.6
1.63
4X
44
2781
IRTS(5)
44
2742
2743.9
80.5
1.40
4Y
41
2584
IRTS(1)
41
2548
2550.0
160.6
1.39
5X
62
3679
IRTS(5)
5Y
57
3353
IRTS(7)
6X
19
1160
IRTS(1)
6Y
17
1076
IRTS(5)
7X
35
1990
7Y
32
1886
8X
29
1990
8Y
27
1835
9X
44
9Y
41
3672
3672.7
575.9
0.19
56
3297
3298.0
495.8
1.67
18
1138
1139.8
117.1
1.90
17
1058
1058.2
170.8
1.67
IRTS(7)
34
1952
1953.6
852.4
1.91
IRTS(7)
30
1820
1820.9
1193.0
3.50
IRTS(6)
29
1946
1946.6
1015.3
2.21
IRTS(3)
27
1805
1806.6
867.3
1.63
2781
IRTS(5)
44
2742
2743.9
1241.2
1.40
2584
IRTS(1)
41
2548
2550.0
1522.9
1.39
62
3679
IRTS(5)
61
3672
3672.7
1761.9
0.19
57
3353
IRTS(7)
56
3297
3298.0
1897.4
1.67
11X
29
3856
IRTS(1)
29
3633
3639.9
953.2
5.78
11Y
28
3309
RTS, IRTS(2)
28
3178
3181.5
1058.7
3.96
12X
32
2475
IRTS(1)
32
2426
2427.0
1074.8
1.98
12Y
32
2233
RTS, IRTS(3)
31
2167
2168.1
1219.0
2.96
13X
29
3856
IRTS(1)
29
3633
3639.9
1538.5
5.78
13Y
28
3309
RTS, IRTS(2)
28
3178
3181.5
1278.8
3.96
10Y
ED
PT
CE
10X
M
61
AC
AN US
CR IP T
Instance
46
ACCEPTED MANUSCRIPT 14X
32
2475
IRTS(1)
32
2426
2427.0
1097.6
1.98
14Y
32
2233
RTS, IRTS(3)
31
2167
2168.1
1154.3
2.96
2384.43
2386.20
814.7
2.28
2442.93
AC
CE
PT
ED
M
AN US
CR IP T
Average
47
ACCEPTED MANUSCRIPT Table B.6: Computational results for VRPDPD (CMT Set 3) Best known
CVNS
k
BKS
Ref.
k
Best sol.
Avg. sol.
Avg. t
Imp(%)
1X
35
1943
RTS
32
1827
1829.2
36.2
5.97
1Y
35
1968
RTS
31
1841
1842.7
72.4
6.45
2X
60
3217
RTS
48
2742
2743.9
549.3
14.77
2Y
63
3292
RTS
48
2809
2810.7
495.4
14.67
3X
69
3805
RTS
59
3412
3416.8
269.3
10.33
3Y
64
3728
RTS
58
3425
3425.3
943.8
8.13
4X
105
5738
RTS
87
4933
4939.4
1281.6
14.03
4Y
97
5328
RTS
87
4947
4948.5
1633.9
7.15
5X
147
7554
RTS
5Y
142
7391
RTS
6X
35
1943
RTS
6Y
35
1968
RTS
7X
60
3217
7Y
63
3292
8X
69
3805
8Y
64
3728
9X
105
9Y
97
10X
AN US
CR IP T
Instance
6449
6450.3
1501.7
14.63
118
6568
6577.2
1615.6
11.14
32
1827
1829.2
36.2
5.97
31
1841
1842.7
72.4
6.45
RTS
48
2742
2743.9
549.3
14.77
RTS
48
2809
2810.7
495.4
14.67
RTS
59
3412
3416.8
269.3
10.33
RTS
58
3425
3425.3
943.8
8.13
5738
RTS
87
4933
4939.4
1281.6
14.03
5328
RTS
87
4947
4948.5
1633.9
7.15
147
7554
RTS
119
6449
6450.3
1501.7
14.63
10Y
142
7391
RTS
118
6568
6577.2
1615.6
11.14
11X
77
8608
RTS
68
7427
7439.6
1507.6
13.72
11Y
75
8372
RTS
68
7499
7512.5
1675.2
10.43
12X
68
4523
RTS
61
3950
3952.8
688.5
12.67
12Y
68
4304
RTS
59
3897
3897.8
723.0
9.46
13X
77
8608
RTS
68
7427
7439.6
1507.6
13.72
13Y
75
8372
RTS
68
7499
7512.5
1675.2
10.43
ED
PT
CE
AC
M
119
48
ACCEPTED MANUSCRIPT 14X
68
4523
RTS
61
3950
3952.8
688.5
12.67
14Y
68
4304
RTS
59
3897
3897.8
723.0
9.46
4409.00
4413.33
928.11
10.97
4983.64
AC
CE
PT
ED
M
AN US
CR IP T
Average
49
ACCEPTED MANUSCRIPT Table B.7: Computational results for VRPSPD and VRPDDP (SCA-CON) VRPSDP (Best known)
VRPSPD (CVNS) Avg. t
Imp(%)
Δk
2
Best sol.
Avg. t Imp(%)
3
k BKS
Δk
SCA3-0
4
635.62 Opt.
635.62
4.77
0.00 0
635.62
8.59
0.00 0
SCA3-1
4
697.84 Opt.
697.84
5.76
0.00 0
697.84
8.07
0.00 0
SCA3-2
4
659.34 Opt.
659.34
8.22
0.00 0
659.34
12.33
0.00 0
SCA3-3
4
680.04 Opt.
680.04
6.24
0.00 0
680.04
9.36
0.00 0
SCA3-4
4
690.50 Opt.
690.50
3.97
0.00 0
690.50
7.14
0.00 0
SCA3-5
4
659.91 Opt.
659.91
5.70
0.00 0
659.90
9.69
0.00 0
SCA3-6
4
651.09 Opt.
651.09
5.15
0.00 0
651.09
7.21
0.00 0
SCA3-7
4
659.17 Opt.
659.17
6.67
0.00 0
659.17
9.33
0.00 0
SCA3-8
4
719.48 Opt.
719.48
SCA3-9
4
681.00 Opt.
681.00
SCA8-0
9
961.50 Opt.
961.50
SCA8-1
9
1049.65 Opt.
1049.65
SCA8-2
9
1039.64 Opt.
1039.64
SCA8-3
9
983.34 Opt.
983.34
SCA8-4
9
1065.49 Opt.
SCA8-5
9
1027.08 Opt.
SCA8-6
9
SCA8-7
CR IP T
Best sol.
1
Instance
AN US
Ref.
VRPDDP (CVNS)
0.00 0
719.47
8.93
0.00 0
8.50
0.00 0
681.00
9.63
0.00 0
5.33
0.00 0
961.50
9.38
0.00 0
6.74
0.00 0
1049.65
9.44
0.00 0
5.45
0.00 0
1039.64
11.62
0.00 0
8.39
0.00 0
979.13
52.08
0.43 0
1065.49
6.07
0.00 0
1065.49
10.20
0.00 0
1027.08
5.57
0.00 0
1027.08
13.02
0.00 0
971.82 Opt.
971.82
6.98
0.00 0
969.50
69.31
0.24 0
10
1051.28 Opt.
1051.28
9.77
0.00 0
1051.28
17.58
0.00 0
9
1071.18 Opt.
1071.18
7.06
0.00 0
1071.18
12.71
0.00 0
9
1060.50 Opt.
1060.50
5.82
0.00 0
1057.26 100.79
0.31 0
CON3-0
4
616.52 Opt.
616.52
6.12
0.00 0
616.52
9.18
0.00 0
CON3-1
4
554.47 Opt.
554.47
4.01
0.00 0
554.47
6.01
0.00 0
CON3-2
4
518.00 Opt.
518.00
9.06
0.00 0
518.00
14.50
0.00 0
CON3-3
4
591.19 Opt.
591.19
6.90
0.00 0
591.19
11.04
0.00 0
CON3-4
4
588.79 Opt.
588.79
3.12
0.00 0
588.79
4.37
0.00 0
CON3-5
4
563.70 Opt.
563.70
6.17
0.00 0
563.70
9.26
0.00 0
AC
SCA8-9
ED
PT
CE
SCA8-8
M
4.96
50
4
ACCEPTED MANUSCRIPT 4
499.05 Opt.
499.05
9.39
0.00 0
499.05
13.15
0.00 0
CON3-7
4
576.48 Opt.
576.48
4.69
0.00 0
576.48
8.43
0.00 0
CON3-8
4
523.05 Opt.
523.05
3.89
0.00 0
523.05
6.61
0.00 0
CON3-9
4
578.25 Opt.
578.25
5.70
0.00 0
578.25
9.12
0.00 0
CON8-0
9
857.17 Opt.
857.17
4.86
0.00 0
857.12
96.91
0.01 0
CON8-1
9
740.85 Opt.
740.85
6.77
0.00 0
739.44 110.83
0.19 0
CON8-2
9
712.89 Opt.
712.89
3.83
0.00 0
706.51
52.36
0.89 0
CON8-3
9
811.07 Opt.
811.07
6.49
0.00 0
811.07
10.82
0.00 0
CON8-4
9
772.25 Opt.
772.25
6.70
0.00 0
771.30
40.25
0.12 0
CON8-5
9
754.88 Opt.
754.88
6.31
0.00 0
754.88
7.81
0.00 0
CON8-6
9
678.92 Opt.
678.92
4.80
0.00 0
678.92
6.10
0.00 0
CON8-7
9
811.96 Opt.
811.96
CON8-8
9
767.53 Opt.
767.53
CON8-9
9
809.00 Opt.
809.00
758.54
758.54
AN US 9.22
0.00 0
811.96
11.40
0.00 0
4.93
0.00 0
766.99
31.09
0.07 0
6.47
0.00 0
797.69 102.94
1.40 0
6.16
0.00
757.78
0.01
M
Average
CR IP T
CON3-6
23.96
AC
CE
PT
ED
Imp(%)1: Percentage improvement between BKS and the CVNS (VRPSPD); Δk2: Deviation in the number of total vehicle (VRPSPD); Imp(%)3: Percentage improvement between VRPSPD and the VRPDDP; Δk4: Deviation in the number of total vehicle (VRPSPD vs VRPDDP); Bold values indicate that CVNS found the best known or the new best solutions. Opt.: Optimal results found in Subramanian, Uchoa [57]
51
ACCEPTED MANUSCRIPT Table B.8: Computational results for VRPSPD and VRPDDP (MON-GAL) VRPSDP (Best known)
Best sol.
r101
12
1009.95 Opt.
2
1009.95
33.86
0.00
0
1009.95
67.04
0.00
0
r201
3
666.20 Opt.
1
666.20
30.56
0.00
0
666.20
59.58
0.00
0
c101
16
1220.18 Opt.
2
1220.18
15.63
0.00
0
1220.18
26.42
0.00
0
c201
5
662.07 Opt.
1
662.07
15.55
0.00
0
662.07
30.78
0.00
0
rc101
10
1059.32 Opt.
2
1059.32
18.25
0.00
0
1058.94
33.39
0.04
0
rc201
3
672.92 Opt.
1
672.92
18.86
0.00
0
672.92
32.62
0.00
0
r121
23
3360.02 370.83
-0.07
0
3355.12
1780.00
0.08
0
r221
5
1665.58 473.59
0.00
0
1665.58
1401.84
0.00
0
c121
28
3629.89
P-ILSRVND
3637.12 917.43
-0.20
0
3630.12
4082.55
-0.01
0
c221
9
1726.59
GENLIS. PILS-RVND
1726.59 398.38
0.00
0
1726.59
1123.42
0.00
0
rc121
23
3306.00
P-ILSRVND
3306.00 485.05
0.00
0
3304.87
2168.18
0.03
0
rc221
5
1560.00 393.51
0.00
0
1560.00
1074.29
0.00
0
1711.05
990.01
0.01
1560.00 Opt. 1711.36
1
1
ED
1665.58 Opt.
Avg. t
Imp(%)
1712.16 264.29
-0.02
Avg. t
Imp(%)
CR IP T
Δk
3357.64 GENLIS
Best sol.
AN US
k
M
Ref.
VRPDDP (CVNS)
Ins.
Avg.
BKS
VRPSPD (CVNS)
AC
CE
PT
Ins.: Test instance; Avg.: Average results; Opt.1: Optimal results found in Subramanian, Uchoa [57]; Opt.2: Optimal results found in Subramanian, Uchoa [58]
52
Δk