A parallel variable neighborhood search for the vehicle routing problem with divisible deliveries and pickups

A parallel variable neighborhood search for the vehicle routing problem with divisible deliveries and pickups

Accepted Manuscript A Parallel Variable Neighborhood Search for the Vehicle Routing Problem with Divisible Deliveries and Pickups Olcay POLAT PII: DO...

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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,

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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.

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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

iV jV

x iV

ij

x iV

ji

R iV

ij

 1 j  L  B

(3)

 1 j  L  B

(4)

 q j   R ji

(5)

j  L

iV

17

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R  R ij

j  B

ji

Ri 0  0

(7)

P q  P ij

iV

j

ij



iL  B

iV

ji

j  B

(8)

j  L

(9)

P0i  0

Rij  Pij  Cxij C

ji

iV

P  P iV

(6)

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iL  B

iV



iL  B

i, j V

x0i   qi iL

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iV

xii  0 i V

x( ni )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









18

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References

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ACCEPTED MANUSCRIPT *29+ Hansen P, Mladenovid N, Pérez JM. Developments of variable neighborhood search. Annals of Operations Research. 2010;175(1):367-407. [30] Bräysy O. A Reactive Variable Neighborhood Search for the Vehicle-Routing Problem with Time Windows. INFORMS Journal on Computing. 2003;15(4):347-68. [31] Polacek M, Hartl RF, Doerner K, Reimann M. A Variable Neighborhood Search for the Multi Depot Vehicle Routing Problem with Time Windows. Journal of Heuristics. 2004;10(6):613-27.

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[32] Polat O, Günther H-O, Kulak O. The feeder network design problem: Application to container services in the Black Sea region. Maritime Economics & Logistics. 2014;16(3):343-69. [33] Avci M, Topaloglu S. An adaptive local search algorithm for vehicle routing problem with simultaneous and mixed pickups and deliveries. Computers & Industrial Engineering. 2015;83:15-29.

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[34] Polat O, Kalayci CB, Kulak O, Günther H-O. A perturbation based variable neighborhood search heuristic for solving the Vehicle Routing Problem with Simultaneous Pickup and Delivery with Time Limit. European Journal of Operational Research. 2015;242(2):369-82. [35] Polacek M, Benkner S, Doerner K, Hartl R. A Cooperative and Adaptive Variable Neighborhood Search for the Multi Depot Vehicle Routing Problem with Time Windows. BuR - Business Research. 2008;1(2):207-18.

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[36] Blum C, Puchinger J, Raidl GR, Roli A. Hybrid metaheuristics in combinatorial optimization: A survey. Applied Soft Computing. 2011;11(6):4135-51.

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*37+ Hansen P, Mladenovid N. Variable neighborhood search: principles and applications. European Journal of Operational Research. 2001;130(3):449-67. [38] Mladenovid N, Hansen P. Variable neighborhood search. Computers & Operations Research. 1997;24:1097-100.

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[39] Kalayci CB, Polat O, Gupta SM. A hybrid genetic algorithm for sequence-dependent disassembly line balancing problem. Annals of Operations Research. 2016:1-34.

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[40] Polat O, Kalayci CB, Mutlu Ö, Gupta SM. A two-phase variable neighbourhood search algorithm for assembly line worker assignment and balancing problem type-II: an industrial case study. International Journal of Production Research. 2016;54(3):722-41. [41] Duarte A, Pantrigo J, Pardo E, Mladenovic N. Multi-objective variable neighborhood search: an application to combinatorial optimization problems. Journal of Global Optimization. 2015;63(3):51536. [42] Clarke G, Wright JW. Scheduling of vehicles from a central depot to a number of delivery points. Operations Research. 1964;12(4):568-81. [43] Doyuran T, Catay B. A robust enhancement to the Clarke-Wright savings algorithm. Journal of The Operational Research Society. 2011;62(1):223-31.

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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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[53] Zachariadis EE, Tarantilis CD, Kiranoudis CT. An adaptive memory methodology for the vehicle routing problem with simultaneous pick-ups and deliveries. European Journal of Operational Research. 2010;202(2):401-11.

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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)

74

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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

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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

26

ACCEPTED MANUSCRIPT Figure 8: Improvements achieved on the total distance

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

CE

863

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ACCEPTED MANUSCRIPT 13X

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.

28

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