Order matters – A Variable Neighborhood Search for the Swap-Body Vehicle Routing Problem

Accepted Manuscript

Order matters – A Variable Neighborhood Search for the Swap-Body Vehicle Routing Problem Sandra Huber, Martin Josef Geiger PII: DOI: Reference:

S0377-2217(17)30393-4 10.1016/j.ejor.2017.04.046 EOR 14412

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

8 March 2016 19 April 2017 20 April 2017

Please cite this article as: Sandra Huber, Martin Josef Geiger, Order matters – A Variable Neighborhood Search for the Swap-Body Vehicle Routing Problem, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.04.046

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Highlights • We study the effects of neighborhood operators for the Swap-Body Vehicle Routing Problem. • An experimental setting is proposed to identify promising orders in a Variable Neighborhood Search.

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• Extensive numerical results are conducted on test instances. We showed that our ideas also perform competitive on additional data sets.

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Order matters – A Variable Neighborhood Search for the Swap-Body Vehicle Routing Problem Sandra Huber∗, Martin Josef Geiger∗

Abstract

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This article presents investigations on the performance of standard and problem specific neighborhood operators for the Swap-Body Vehicle Routing Problem. More specifically, our work analyzes the contribution of each operator to the solution quality. In addition, an experimental setting is defined that can be utilized to identify promising sequences in a Variable Neighborhood Search. The experiments verify that the sequence matters since two best known solutions can be equaled and 16 out of 18 solutions can be improved, with a maximal improvement of 2.25% and an average improvement of 0.70%.

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Key words: Vehicle Routing Problem, VeRoLog Solver Challenge, Variable Neighborhood Search, Order of Neighborhood Operators, Parallel heuristic

Introduction

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The Swap-Body Vehicle Routing Problem (SB-VRP) [23] is a challenging generalization of the Capacitated Vehicle Routing Problem (CVRP) where additional decisions must be made, such as the selection of the vehicle configuration or of swap locations. Research in this direction was stipulated by [1, 26, 27, 33, 34, 44, 53]. In the SB-VRP customers are serviced by a truck or a train whereas due to some practical constraints, some customers can only be visited by a truck. The other customers can either be delivered by a truck/ train. Similar to this problem domain is the truck and trailer routing problem (TTRP) which recently received higher research attention [11, 12, 13, 31, 35, 42, 50, 52]. A literature overview of problems related to the SB-VRP is given in [26, 34]. The terms ‘train’ and ‘trailer’ can be used interchangeably. Both descriptions mean that a trailer is attached to a truck. In order to tackle the real-life SB-VRP a parallel Variable Neighborhood Search is implemented. Moreover, the aim is to study the effects of the proposed neighborhood operators on the solution quality. This aspect is decisive when the computational time is scarce and a decision must be made which operators should be included in the sequence. In addition, a structured process is developed to identify how promising sequences of the neighborhood operators can be established. This detailed investigation is crucial since often rules of thumb are used [41] to obtain the sequences of the neighborhood operators. An literature overview on the determination of a sequence is presented in Section 2. Based on these research questions, an experimental setting is defined that evaluates the performance of the neighborhood operators. Extensive computational experiments are carried out on benchmark instances. Our results show that two best known solutions can be equaled and 16 out of 18 instance can be improved, with a maximal improvement of 2.25% and an average improvement of 0.70%. This paper is organized as follows. Section 2 discusses related literature on the identification of good sequences for the Variable Neighborhood Search (VNS). The problem statement of the SB-VRP is described in Section 3. In Section 4 the parallel

∗ Helmut-Schmidt University, University of the Federal Armed Forces, Holstenhofweg 85, 22043 Hamburg, [email protected]

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VNS is outlined, followed by the computational results in Section 5. Finally, Section 6 concludes the work.

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Literature Review

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In the basic VNS an improvement procedure, a shaking move and a neighborhood change step is alternately applied, until a predetermined termination criterion is reached. A shaking move is used to overcome local optima traps. The improvement procedure can be a local search with simply one neighborhood operator or with several neighborhood structures [37]. Thereby, a finite set of selected neighborhood structures with Nk , k = 1, . . . , kmax are defined. When kmax > 1 some questions must be answered: 1) what Nk should be used and how many of them? 2) what sequence should be applied? and 3) what strategy should be used in changing neighborhoods [19, 22, 36]? Our emphasis lies in the review of the first and the second question. In particular, we are interested in the process of verifying promising sequences and how the contribution of each neighborhood operator is determined. This is an important research question since it is not obvious how the order of the neighborhood operators should look like. Often rules of thumb are used to identify a meaningful order which could be i) first search smaller neighborhoods, ii) search neighborhoods first for which fast algorithms exist and iii) apply “promising” operators first [41]. The strategy how to change the neighborhoods is also a crucial part of a VNS, which goes beyond the scope of this paper and must be defined as an separate research question. An overview of the literature is presented in Table 1 with respect to the used neighborhood operators and the order of these operators for the Traveling Salesman Problem (TSP), the Vehicle Routing Problem (VRP) and variants of the VRP. Note that we used the description of the neighborhood operators given in each study. The work of [3] investigates the performance of various popular neighborhood operators such as 2-Opt, 3-Opt etc. in a Reactive Tabu Search and Variable Neighborhood Search framework for the TSP with time windows (TSPTW). Their goal is to analyze the effects on the solution quality, when the same operators are applied in different frameworks. However, the investigation of the sequence plays a minor role. Local search operators are evaluated for the multiple Traveling Salesmen Problem (mTSP). Several orders are considered and one good order is identified for further tests. It is shown via computational experiments that the order has an impact on the performance of the algorithm [46]. The k-dissimilar Vehicle Routing Problem or kd-VRP is studied by [49]. Different sequences of operators are tested and the most promising order is used for further experiments. It would be interesting to know how the experimental setting is configured since no numerical results are presented. In the study of [45], the Split Delivery Vehicle Routing Problem (SDVRP) is analyzed for an iterated local search heuristic. A Randomized Variable Neighborhood Descent (RVND) is implemented and it is mentioned that every operator plays an important role. However, no experiments are reported to support this finding. In addition it is stated that the effects of the operators depend on the properties of the input data. For example, the problem specific operators (see Table 1) are more important, when the average demand is high, since splitting the demand appears more often. This observation suggests to apply different sequences depending on the characteristics of the benchmark instances. Nevertheless, these considerations are not used in the experimental investigation. One deterministic order and a random ordering of the proposed operators is tested for the Heterogeneous Fleet Vehicle Routing Problem (HFVRP) [39]. It is evaluated that the random ordering outperforms all variants. The authors do not report how the sequence of the operators is determined and in addition, only one sequence is tested. For the Vehicle Routing Problem with Simultaneous Pickup and Delivery (VRPSPD) a RVND is performed [47]. Before, a similar solution approach with a deterministic order is implemented [48] and it is concluded that the RVND approach

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Table 1: Overview on the investigated local search operators and the ideas for applying a promising order of the operators Problem

Local search operators

Investigation on the order of the operators

TSPTW [3]

2-Opt, Or-Opt, 3-Opt, 1-shift, The determination of the order is not Node Exchange clear.

1-PDTSP [54] Swap, Insert, Exchange, Back in- One deterministic sequence is used. It sert, 2-Opt, 3-Opt, 3-Opt with is no information given how the sequence is determined. change direction One-point move, Or-Opt, Two- Experimental tests are used to deterpoint move, Three-point move, 2- mine the sequence of the operators. Opt

kd-VRP [49]

Internal Or-Opt, Internal Relo- Experiments with several configuracate, Internal 2-Opt, External Ex- tions and identification of the most change, External Relocate, Ex- promising order. ternal 2-Opt, External CrossExchange

SDVRP [45]

Inter-route: Shift (1,0), Swap A random approach is assumed where (1,1), Shift(2,0), Swap(2,1), no order must be specified. Swap(2,2), Cross Problem specific: RouteAddition, k-Split, Swap(1,1)∗ , Swap(2,1)∗

HFVRP [39]

Inter-route: Shift (1,0), Swap A random ordering is compared with (1,1), Shift(2,0), Swap(2,1), one deterministic sequence. Swap(2,2), k-Shift, Reinsertion, Or-Opt2, Or-Opt3, 2-Opt, Exchange

VRPSPD [48, 47]

Inter-route: Shift (1,0), Swap A deterministic and a random order(1,1), Shift(2,0), Swap(2,1), ing is applied. Swap(2,2), Cross Intra-route: Or-Opt, 2-Opt, Exchange, Reverse

VRP with Backhauls [7]

Intra-route and inter-route cus- A deterministic order is defined, but tomer relocation, Intra-route and no information is given how the order inter-route customer exchange, is identified. Inter-route crossover, Intra-route 2-Opt

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mTSP [46]

In total 9 operators are applied in- One order is identified without decluding e. g. Swap and Exchange scribing, why this sequence is semoves. lected.

Truck and Trailer RP [10]

2-Opt, 2-Opt∗ ,Exchange, Relocate The sequence of the operators is deProblem specific: Relocate-sub- scribed. However, no further informatour, Switch-vehicle-type tion is given why especially this ordering is promising.

VRPM [2]

Several destruction and insertion A removal and a insertion operator is operators are defined. selected for each iteration. An operator is chosen by taking into account the previous outcomes.

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VRP with multiple trips [6]

SRSPDTW [9] CROSS, 2–opt∗ , relocation, reloca- A study for a promising neighborhood tionChanging and swapInter oper- structure is established for a represenator tative subset of instances. SB-VRP [8]

In total 42 operators are tested and An automated tuning tool is used to clustered. analyze several scenarios.

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achieves better results. Again, the question arises how the sequence of the operators is determined. In [2] several destruction and insertion operators are proposed for the Vehicle Routing Problem with multiple routes (VRPM). The idea is to establish promising operators throughout the search and select them. Thereby weights are introduced and updated during the search based on the performance of the neighborhood operator. The work of [9] explores several operators in the ship routing and scheduling problem with discretized time windows (SRSPDTW). A good sequence is determined by analyzing the contribution of each operator with respect to the solution quality as well as computational times. In addition to the previous investigations, some VNS studies in different problem areas are highlighted. The authors in [41] determine the neighborhood structures dynamically during the search for the well-studied multidimensional Knapsack Problem (MKP). Thereby, improvement-potentials are identified by quickly solving a Linear Programming (LP) relaxation of an Integer Linear Programming (ILP) formulation with each neighborhood operator. Then, the operators are sorted in a decreasing order with respect to the computed improvement-potential. This seems to be a promising idea. However, in some cases it might be difficult to find a relaxation of an investigated problem that can be quickly solved. This is especially important when the computational time is scarce. Extensive studies have been applied for the single machine total weighted tardiness problem to investigate local search neighborhood operators in a VNS. It has been illustrated that the sequence of the neighborhood operators influences the outcome and findings on the relative performance of the proposed operators have been reported [14]. In previous studies on the VRP limited effort has been put in finding promising orders for the neighborhood operators. In many cases, it is not reported how the experimental setting is defined to specify promising orders (cf. [3, 6, 7, 10, 25, 38, 43, 45, 54]). However, several studies [2, 14, 41, 46] propose that the order has an influence on the performance of the algorithm and include this aspect in their research. Based on the overall findings, we identify a research gap and propose a structured way for testing the influence of different sequences on the solution quality (see Section 5).

Problem statement

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The Swap-Body Vehicle Routing Problem was announced by the Working Group on Vehicle Routing and Logistics Optimization (VeRoLog)1 [23] within EURO, the Association of the European Operational Research Societies in collaboration with the German company PTV Group 2 . Due to the recent introduction of the problem, only few studies exist within its scope [1, 26, 33, 34, 51, 53]. Nevertheless, the problem has a high relevance for many industrial applications. Also a variant, namely the bi-objective Swap-body Inventory Routing Problem is analyzed in [27]. The Truck and Trailer Routing Problem (TTRP) [10] is similar to the SB-VRP. However, e. g. more actions, such as a swap and an exchange action (see Figure 2), can be applied at swap locations [1, 26]. Thus, more complicated routes are possible [1]. In addition, more terms are incorporated in the objective function, such as fixed usage cost for the truck and the trailer [26]. The main characteristics of the SB-VRP and the TTRP are summarized in Table 2. The single-objective function F of the SB-VRP tries to minimize the total costs of the routes which are the sum of the following components: If a truck is used in a solution, then fixed costs [M U/usage] must be taken into account. Variable costs consist of the total sum of traveled distances [M U/km] and the total sum of driver costs [M U/h]. Additional variable costs [M U/km] and additional fixed costs 1 “The purpose of VeRoLog is to constitute an aggregation and reference point for the rich and active research community in order to e. g. actively promote the interest on Vehicle Routing.” (see www.verolog.eu) 2 www.ptvgroup.com This company has its headquarters in Karlsruhe, Germany and offers software solutions for transportation of goods and people.

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Table 2: Characteristics of the SB-VRP and the TTRP Characteristics

SB-VRP

Depot

One depot

TTRP [10]

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One depot with a time window Number of vehicles is not Number of vehicles is limited bounded Each swap body has the same Differnet capacities for the capacity truck and the trailer Customers Three types of customers Two types of customers Each customer has a time window Network Asymmetric travel times and Symmetric travel times and distances distances Triangle inequality does not Triangle inequality is valid hold Decoupling point Swap location Vehicle customer No transshipments are Load transfers are applicable possible Park, pickup, swap and Park and pickup operations exchange operation Objective function Fixed and variable costs Minimization of the total tour length

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[M U/usage] arise when the semi-trailer is attached to a truck. Note that no costs are related to the swap bodies. The proposed SB-VRP [23] can be defined on a directed, asymmetric graph G = (V, A), where V is the set of nodes and A is the set of arcs. Three subsets of nodes are comprised by V : the depot node v0 , the subset Vs containing m swap locations and the subset Vc of n customers. Distances for travelling from node i to node j are given by dij and vice versa. Also driving times from node i to node j are given by tij and vice versa. Note that the dij and tij values are calculated based on shortest path computation on a digital road network. Furthermore, due to the origin of the problem from a real industrial setting, the triangular inequality does not hold for the asymmetric dij and tij values [23].

swap body

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Figure 1: Valid vehicle configurations: truck and train. [23].

Possible vehicle configurations are presented in Figure 1, namely a truck moving with one swap body and a train configuration (truck and semi-trailer) carrying two swap bodies SBl . The loading capacity for each SB1 and SB2 is Q. Thus, the train has a maximum load of 2Q. An unlimited fleet is stationed at the depot v0 . Besides, every vehicle configuration starts and ends at the depot. When returning to the depot, a truck/train must have the identical swap bodies with which it had departed. The distinction between a train and truck configuration is made since the set of customers is divided into 1) truck customers which can only be visited by a truck, 2) train customers that can be reached by truck or by train and 3) mandatory train customers which must be visited by a train since the demand qi > Q. From a practical point of view, these differences are due to accessibility constraints (e. g. limited manoeuvering space) [13]. Each customer i has a demand qi ≥ 0, which must either be fulfilled by one truck or one train. Note that it is not permitted to transfer load partially or

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Figure 2: Exemplarily description of the possible actions at the swap location [23].

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completely to another SB. In addition to the capacitated VRP, swap locations are introduced which can be used by the train combination. In those plans, a train may change its configuration by performing an action at the swap location. An overview of the four possible actions are presented in Figure 2, such as park, pickup, swap and exchange. 1. In particular, a train can park the semi-trailer with SB2 and then the truck continues only with SB1 . 2. A pickup action of SB2 must take place at some point in time since the restriction is given that the train configuration must return to the depot. Thereby, the initial and finial position of the swap bodies can be ignored.

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3. Regarding the swap action, the truck parks the currently carried SB1 and continues with SB2 , which was parked before at the swap location.

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4. The exchange operation parks the semi-trailer and exchanges the swap bodies so that the truck moves with SB2 .

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During the planning horizon an unlimited number of vehicles and an unlimited number of actions can appear at each swap location. Besides, no detailed scheduling must be determined for the swap locations, e. i. neither capacity constraints nor waiting times are evident at the swap location. With respect to the routing of a train, the number of used swap location is not bounded. It must be mentioned that every driver has a maximum driving time T which may not be exceeded. T includes the service time si for each customer i, the driving times tij and the handling times for each action at the swap location. Different handling times tpark , tpickup , tswap , texchange are defined for each action. It is prior known that e. g. in a typical data set a park action is less time consuming than an exchange operation. In the literature, a mixed integer programming formulation for the SB-VRP is presented by the study of [51].

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Solution method

An alternative is represented by several tours whereas a single tour has several segments. In the general case, the number of segments is not limited. Taking into account that a vehicle moves through the network and visits customers, at some point, the vehicle cannot visit more customers due to time- and quantity restrictions (see Figure 6). Since this problem is inspired by a practical application, the driving time cannot be extended due to (inter) national regulations. With respect to these aforementioned constraints, an alternative is represented by at most four segments Sk with k = 1, . . . , 4. Implementing this alternative representation comes with a reduced search space. However, numerical results in Table 3 gives that both, the average

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Table 3: Average time- and vehicle-utilization for Table 4: Utilization values for the the instances small normal instance Avg. timeutilized %

92.50 93.04 92.83

83.30 89.58 85.62

91.35 93.00 94.85

86.93 87.42 88.29

85.76 89.12 87.28

97.54 91.69 94.01

93.05 92.89 92.23

92.43 91.47 91.43

97.15 90.05 97.13

64.45 70.79 65.89

93.79 95.61 92.82

95.22 89.79 94.63

Identifier VehicleTimeof the route utilization % utilized % 1 2 3 4 5 6 7 8 9 10 11 12

98.00 61.00 95.00 92.50 95.00 96.00 98.50 99.50 96.50 96.00 96.00 90.00

97.01 99.34 99.83 53.89 85.76 95.83 92.33 36.46 89.31 95.24 95.56 86.88

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small all with trailer all without trailer normal medium all with trailer all without trailer normal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal

Avg. vehicleutilization %

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vehicle- as well as time-utilization, are relatively high for each instance (a detailed explanation of the instances is presented in Section 5.1). This shows that including an exchange operation at the swap location is difficult since the time and/ or the quantity restriction is not be respected. In Table 4 are exemplarily given the timeand vehicle-utilization for each tour. Here, we have in total twelve routes. The second tour e. g. uses 99.34 % of the driving time, but only 61% of the vehicle. Three different tour types with a different number of segments are shown in Figure 3. One possibility is a truck tour where only one segment S1 is used and S2 = ∅ ∧ S3 = ∅ ∧ S4 = ∅. When a train configuration leaves the depot, two options are possible. On the one hand, a train tour can include three segments S1 , S2 , S4 and S3 = ∅. On the other hand, customers can be allocated to all four segments. In this regard, segments S2 and S3 can be seen as subtours where only one swap body SB2 respectively SB3 is utilized. Introducing this representation comes with several advantages:

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• When S2 = ∅, inserting customers in S3 or S4 does not make sense and is, consequently, not checked. Thus, the number of possible moves is reduced. • Likewise, when S2 = S3 = ∅, there cannot be an insertion in S4 .

• Insertions of customers can be quickly checked by removing one arc and replacing it with two others, which is well-known. Our tailored data structure with fixed segments allows us to efficiently implement a special case: Whenever the first customer is inserted in S2 , then one (the last) arc in S1 points to the swap location, and one arc in S4 , i. e. the one pointing from the swap location to the depot, is introduced. As a corollary, all possible insertion and removal moves can be checked in constant time.

• In conclusion, significant speedups are obtained by the dedicated representation, and we believe that such elements contribute to the overall success of our method, as we have seen in other studies [15].

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2. Train tour with three segments

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Depot S1

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Mandatory train customer Truck configura"on

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Figure 3: Illustrative example of the tour types with different number of segments.

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A classification of each segment is given with respect to allowed customers in Table 5. For example, it is presented that train customers as well as mandatory train customers are allowed on segment S1 respectively S4 . However, it is not possible to allocate a truck customer on these segments. To the segments S2 respectively S3 truck and train customers can be included, but no mandatory train customers (qi > Q). Table 5: Classification of the four segments, when a train configuration leaves the depot Mandatory train customer

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S1 S2 S3 S4

Train customer

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Additionally, it is clear which action must occur when S2 is assigned with customers, such as a park action. Alternatively, when also customers are positioned in S3 a swap action is necessary. Moreover, the solution always includes the information which SB is used to satisfy the order of the customers in a train configuration. This can be easily checked for the segments Si which is given in Table 6. For the segments S2 and S3 the sum of the demands qi must be smaller or equal to the capacity Q of SB1 respectively SB2 . Furthermore, the sum of all fulfilled requests on the four segments must be smaller/ equal to 2Q (qik is the delivered quantity of customer i on segment k). The demand of the customers on S1 as well as on S4 can then either be delivered by SB1 respectively SB2 or by both swap bodies.

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Table 6: Quantity constraints, when a train configuration leaves the depot Type of segment

Quantity check for every Sk

Overall Quantity check

S1 S2 S3 S4

–P Pi∈S2 qi ≤ Q i∈S3 qi ≤ Q –

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Construction phase

In order to obtain an initial solution, a customer is randomly selected. With respect to the vehicle configuration, a truck or a train is chosen at random to initiate a tour.

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As an assignment criteria, we use a priority rule for the segments Sk . The algorithm tries to assign the customer to the first, second etc. segment Sk (train configuration). Using a truck configuration implies that customers only utilize S1 . If the customer does not fit in any Sk , a new tour is opened. This occurs when the maximal driving time T or delivery amounts for the different segments are exceeded. Our procedure is applied until all customers are assigned to segments. Note that the prioritization of the second or third route segment over the first one does not improve the solution quality. This is due to the improvement procedure which puts the customers back on the first segment.

4.2

Improvement phase

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The initial route plan Sl is improved by developing a VNS [5, 36]. The general procedure of the VNS is shown in Algorithm 1. As an input the number of tours to remove and the set of neighborhood operators is needed. The routine starts with an already constructed solution of phase 1, including a run of the neighborhood search. After that, Sl is improved by alternately executing the perturbation (Algorithm 2) and the local search (lines 5 – 10 of Algorithm 1) until the imposed CPU time is reached. Perturbation [32] and shaking [21] are similar strategies to diversify the search. In a VNS, the shaking procedure often generates a solution at random from a neighborhood [21, 20]. Differently to this shaking procedure, we apply a stronger perturbation by removing entire tours in order to avoid falling back to the recently visited local optimum [32]. Such perturbations are e. g. applied in Iterated Local Search [32]. Numerical results of the effect of the perturbation are given in Section 5.5. Local optima traps are resolved by a perturbation. As shown in Algorithm 2, the alternative is ruined by randomly removing r entire tours. In algorithm 3 the reconstructing steps are presented. The InsertCustomer procedure requires as an input the removed customers of the perturbation (OpenCustomers). Our procedure firstly chooses a customer at random. After that, it is tried to insert the customer in an existing tour. If this step is not successful, either a truck or a train tour is initialized and the customer is assigned to Sk .

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Sl ← InitialSolution repeat 0 Sl ← Perturbation(Sl, r) h←1 while there is an improvement do while h ≤ hmax do 00 0 Sl ← Nh (Sl ) h ← h+1 00 if Sl better than Sl then 00 Sl ← Sl

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Algorithm 1: Variable Neighborhood Search Input: The set of neighborhood operators Nh , for h = 1, . . . , hmax ; number r of tours to remove

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until Termination criterion is met

In the local search phase (lines 5 – 10 of Algorithm 1) hmax neighborhood structures are executed sequentially. Thereby, the sequence can e g. be determined a priori or selected randomly (see Section 5.3). The operators are applied until no improvement is achieved. A promising order of the neighborhood operators is investigated in Section 5. Based on the literature overview in Section 2, we identified that several inter-tourand intra-tour operators are commonly used for the local search. In addition, often a problem specific operator is utilized [45]. With respect to these findings, the following

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Algorithm 2: Perturbation(Sl, r)

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for m = 1 to r do Choose a tour in Sl at random and remove the tour Add the customers to OpenCustomers

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InsertCustomer(OpenCustomers) in Sl

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repeat Choose a customer in OpenCustomers at random Customer is Inserted ← False for u = 1 to NumberOfTours do if Customer is assigned to Sk then Customer is Inserted = True exit for if Customer is Inserted = False then Create a new truck xor train tour at random Assign customer to Sk

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Algorithm 3: InsertCustomer(OpenCustomers)

neighborhood operators are implemented. The acceptance criterion of every operator is the same: a neighboring alternative is accepted iff F (Sl00 ) < F (Sl) ∧ Sl00 ∈ X with X denoting the set of feasible alternatives. 4.2.1

Problem-Specific-Operator

4.2.2

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Change swap locations (CSL) modifies the used swap location of every tour (O(Rm) with R being the number of routes and m being the number of swap locations). It is checked whether a different swap location is better suited for the identified tour. This is necessary since the initially constructed tour assigned the best swap location of the customer that is assigned first. Intra-Tour-Operator

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Intra-Move (INTRA) relocates a selected customer in another Sk in the same tour. This procedure is exemplarily presented in Figure 4 where the train customer of S1 in Tour 1 is inserted in S2 before the truck customer (tour 1a) and after the truck customer (tour 1b). In tour 1c the train customer of S1 is inserted in S4 before respectively after the train customer (O(n2 )). The classical 2-opt (2OPT) tries to improve a single tour by replacing 2 arcs with 2 other arcs. This replacement of arcs is applied to every segment when the number of customers is greater than 1. Since the travel times as well as the distance times are asymmetric, we have to evaluate the whole tour in terms of the travel times and the distance cost. Note that an overall evaluation is necessary since the direction can change. The complexity is O(n2 ) [4]. Our 3-opt (3OPT) is replacing 3 arcs with 3 other arcs (O(n3 )) [4, 30]. Similar to the 2OPT, the idea is computed for every segment when the number of customers on the segment is greater than 3. The overall evaluation of the tour must be computed since the direction is important in the asymmetric case. 4.2.3

Inter-Tour-Operator

Two-Inter-Exchange (2EX) is a move in which the assignment of 2 customers are swapped [17]. One example is presented in Figure 5 where two tours are selected. Then, the train customer of S1 in tour 1 is inserted in S1 of tour 2 (new tour 2) and a mandatory train customer is moved to S1 in tour 1 (new tour 1). Note that

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Figure 4: Explanation of the INTRA operator.

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the procedure tries every possible position in the tour and uses the best insertion position. It is e. g. tested to move the mandatory train customer to S4 . Due to quantity constraints the customer cannot be assigned to S2 . The complexity is O(n). Here, only tours are tested that are ‘close’ to the selected customer. Close tours are identified by the neighboring customers of the chosen customer (measurement is distance). The value is set to 100. New Tour 1

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Figure 5: Explanation of the 2EX operator.

Three-Inter-Exchange (3EX) switches the assignment of 3 customers which is similar to an ejection chain procedure [16]. Customer v1 is inserted at the best position in tour 2. The second customer v2 is moved to the third customer’s tour and the third is put on the best position in the tour of the first customer. This idea is illustrated in Figure 6. The complexity is O(n). Similar to the 2EX, we determine close tours and restrict the number of neighboring customers to 100. Inter-Move (INTER) removes a chosen customer of a selected tour and tries to position the customer in each segment in another tour. The complexity is O(n2 ) [9].

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ACCEPTED MANUSCRIPT Tour 1

New Tour 1

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Tour 2

New Tour 3

New Tour 2

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Figure 6: Example of the 3EX operator.

4.3

Parallel algorithm

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Parallel and Distributed Computing Systems (PDCS) are popular to solve combinatorial optimization problems and are especially used for larger instances [29]. Applying parallelism to a metaheuristic can allow to reduce computational time when the sequential program is partitioned or can increase the exploration of the search space by the application of independent search threads [40]. Similar to the studies of [29, 47] our parallel algorithm is based on a masterworker approach. Variable Neighborhood Searches with the same parameters are performed in parallel. The local search is executed separately on the four threads (workers). Thereby, the master thread collects the best solutions of each slave and submits the best of them to all slaves. In addition, the master thread archives the global best solution. The communication of the master thread with the workers is determined by a synchronisation time in seconds which depicts when an exchange of information between the master and the workers is performed. For example a value of two seconds means that the search is executed for around two seconds (worker can finish the current VNS) and then the information is exchanged and the global best solution might get updated. Promising values for the synchronisation time are further investigated in Section 5.4.

Computational results

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The VeRoLog Challenge was divided into two phases: 1) pre-selection part and 2) the final phase. Each phase includes so-called hidden and public instances. Data sets small, medium as well as large1 were public during the pre-selection phase and the large2 set was not known before. For the final part, the instances final and final random were used as private test data. With respect to the hardware configuration, the VeRoLog organizers specified that a standard desktop computer with four cores and 16 GB RAM is utilized for 600 seconds to test the programs. After the pre-selection phase of the competition, the organizers verified our results and gave us best known solutions (BKS) for selected test instances which are presented in the second column of Table 9. While we do not know the exact hardware configuration used by the organizers, it can be suspected that it is similar to our configuration since matching results are achieved. We execute R

the algorithm on an Intel Xeon X5650 2.66 GHz with 24.0 GB of RAM. However, the given memory is not fully needed in our experiments. All experiments were fixed with these restrictions and since the algorithm is randomized, 30 runs have been conducted for each instance.

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ACCEPTED MANUSCRIPT Table 7: Main characteristics of the instances provided by the VeRoLog Solver Challenge (k represents thousand) Instance

Type of customers

Demand

Additional characteristics

#train #mandatory #truck Avg. Max. #swap Q customers train customers demand demand locations customers

released in 2014

small all with trailer 56 all without trailer 0 normal 41

1 0 1

0 57 15

149 140 149

995 500 995

20 20 20

500 Feb 500 Feb 500 Feb

medium all with trailer 206 all without trailer 0 normal 186

0 0 0

0 206 20

121 117 121

720 500 720

41 41 41

1k 1k 1k

large1 all with trailer 548 all without trailer 0 normal 498

0 0 0

0 548 50

119 116 119

995 500 995

large2 all with trailer 550 all without trailer 0 normal 500

0 0 0

0 550 50

107 107 107

960 960 960

final all with trailer 549 all without trailer 0 normal 499

0 0 0

0 549 50

108 108 108

960 960 960

final random all with trailer 549 all without trailer 0 normal 499

0 0 0

549 549 50

587 594 604

999 1,000 999

CR IP T 99 99 99

1k 1k 1k

Feb Feb Feb

101 101 101

1k 1k 1k

May May May

102 102 102

1k 1k 1k

July July July

102 102 102

1k 1k 1k

July July July

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5.1

Test instances

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Our algorithm was tested on 18 real-world instances which were provided by VeRoLog and the PTV Group. The instances are available on the vehicle routing problem repository http://www.vrp-rep.org/datasets.html. An overview of the data is presented in Table 7 and classified with respect to the type of the customers, the demand data and some additional characteristics. The latter aspect gives information on the number of swap locations, the capacity of one swap-body (Q) and the release dates of the instances. With the last aspect it is illustrated that during the challenge public and private sets were introduced to evaluate the algorithms. In the final phase of the challenge the unknown private instances final and final random were also used to measure the performance of the algorithm. In total, 6 sets of instances were released, each set comprising 3 variants: normal, all with trailer and all without trailer. The main difference between those variants is the type of customer included in the variant. Normal instances are composed of truck-, mandatory- and train customers, the all with trailer instances include trainand mandatory customers and all without trailer instances only consist of truck customers. Moreover, the sets come mainly in 3 sizes. Small instances containing up to 57 customers, the medium instances consider around 200 customers and the large instances assume around 550 customers (large1, large2, final, final random) [26]. Looking at the order information in more detail, it can be seen that the average demand of the final random instance is more than 5 times higher than for the other instances. The maximal values (max. demand) of the all with trailer and normal instances are the same for each set. However, in the case of the all without trailer instances the maximal value is reduced (small, medium, large1 ) with the exception of the final random instance.

5.2

Mixed integer programming formulation

A mixed integer programming formulation (MIP) for the SB-VRP is proposed

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in [51]. We implemented the MIP in CPLEX 12.7 and modified several expressions. Constraints (1) correct constraints (16) of [51], (2) change constraints (33), and constraints (3) modify (36). S xT i,k,t ≤ xi,k,s + 1 − as,t , i ∈ L, s ∈ S, t ∈ T, k ∈ K S qk,s − S qk,s −

X

i∈C

S zi,k,s −Q

X

i∈C

S zi,k,s −Q

X

i∈L,t∈T

X

i∈L,t∈T

(1)

E T (oW i,k,s,t + oi,k,s,t ) ≤ qk+1,t ≤

E (oW i,k,s,t + oi,k,s,t ), k ∈ K, k ≥ |K| − 1, s ∈ S

S S xS i,k,s + xj,k+1,s − 1 ≤ yi,j,s , i, j ∈ N, s ∈ S, k ∈ K, k ≤ |K| − 1

(2)

(3)

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Table 8: Results of the MIP model for the modified instance small – all with trailer with twenty swap locations. NA means not available within 600 seconds. MIP

Our algorithm (single run)

3 customers 5 customers 10 customers 20 customers 30 customers 40 customers 57 customers

NA NA NA NA NA NA NA

467.44 1,076.25 15,337.57 2,527.28 3,232.98 3,881.44 4,730.70

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Number of customers

Comparison of the VeRoLog Challenge results

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In the study of [51] no insights are provided on the used instances. We tested the small instance with 57 customers and 20 swap-locations. Since no optimal results can be computed for the VeRoLog data set, we modified the number of customers, while keeping the number of swap locations. Results are given in Table 8 and compared to our algorithm. Note that the configuration described in section 5.7 is utilized. The model could not compute results within 600 seconds, which is due to the NP hardness of the problem as well as insufficient memory. E. g. for an instance with 57 customers and twenty swap-locations, the MIP has 990.575 constraints and 39.627 variables. This is in line with the study of [51] who reported that only trivial instances can be solved. However, no numerical results are reported. Alternatively, we reduced the number of swap locations by ten and conducted identical experiments. The MIP solver generated an optimal solution for three customers and ten swap locations. Our algorithm is equally able to compute the optimal solution.

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The comparison of the results on the Challenge instances in Table 9 presents that our algorithm achieves competitive numerical values. In the literature, solutions for the SB-VRP can be found in [1, 33, 34, 51]. To the best of our knowledge, no further studies are published for these instances. Therefore, our results (ranked third in the final phase of the VeRoLog Challenge) are compared to the study ranked second [34] and the work ranked seventh [24, 33]. Additionally, results are presented for our Random VNS. Overall 27 teams participated in the challenge [18]. Our VNS version for the pre-selection phase of the VeRoLog Challenge had the following neighborhood operator sequence: 2EX, INTER, INTRA, CSL and 2OPT. Synchronisation time was set to 3 seconds (described in Section 4 and is further investigated in Section 5.4) and the solution was perturbed by removing two complete routes (see Section 4). Throughout the analysis of the computational results, different measures are applied to evaluate the results. Gap BKS gives the gap between the best solution and the BKS (Gap BKS = ((Best. Sol. - BKS)/BKS)*100) and the average gap BKS (Avg. Gap BKS ) is the gap between the average solution and the BKS (Avg. Gap

Todosijevi´ c et al. [51]

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Absi et al. [1]

Our VNS [26]

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Miranda-Bront et al. [34]

BKS

Our Random VNS

Avg. Gap BKS

BKS

Gap BKS

BKS

Best Sol.

Best Sol.

Avg. Sol.

Avg. Sol.

Avg. Gap BKS

Avg. Gap BKS

Gap BKS

Gap BKS

Best Sol.

Best Sol.

Avg. Sol.

Avg. Sol.

Gap BKS

Gap BKS

Best Sol.

Best Sol.

Avg. Gap BKS

(Our VNS)

Gap BKS

[26, 33, 34]

0.02 1.44 1.02

(organizers)

0.01 0.00 0.22

4,716.58 4,992.44 4,802.38

4,730.92 4,839.64 4,808.17

4,724.81 5,007.74 4,808.18

4,731.81 4,909.25 4,846.31

0.00 1.52 0.43

0.01 0.42 0.16

0.00 0.33 0.00

0.01 0.00 0.15

4,730.63 4,855.62 4,797.55

4,730.92 4,839.64 4,804.97

4,730.63 4,913.09 4,818.15

4,730.92 4,860.06 4,805.32

3.01 10.68 3.37

0.01 8.46 1.04

4,873.05 5,356.36 4,959.00

4,731.02 5,249.18 4,847.63

– – –

-0.12 3.47 0.22

4,730.63 4,839.64 4,797.55

-0.30 3.16 0.10

– – –

0.12 0.24 0.29

1.11 0.78 1.68

7,764.47 8,065.10 7,819.35

0.37 -0.10 0.22

7,827.93 8,112.88 7,876.66

7,784.36 8,037.82 7,813.73

2.61 2.71 3.12

7,841.59 8,108.62 7,927.86

1.18 2.19 2.00

0.82 0.92 1.14

7,847.30 8,221.32 7,952.30

0.00 0.00 0.27

7,957.60 8,263.76 8,040.17

7,755.43 8,045.47 7,817.83

7.48 7.25 6.42

7,818.87 8,119.43 7,885.75

8,335.55 8,628.37 8,297.25

0.13 4.19 0.49

– – –

4.01 3.63 4.04

7,765.75 8,382.80 7,834.78

7,755.43 8,045.47 –

2.05 1.44 2.18

0.93 0.84 1.03

– – 7,796.68

20,630.30 21,560.80 20,983.90

1.99 1.76 2.64

21,026.51 22,026.64 21,365.48

1.08 0.15 1.12

3.55 2.48 2.51

20,432.81 21,287.65 20,766.60

2.79 2.01 1.72

20,618.02 21,630.01 21,077.88

20,778.30 21,683.60 20,889.20

1.32 0.95 1.11

20,932.23 21,782.45 21,050.29

0.00 0.00 -0.05

5.45 5.48 7.38

24,965.10 26,515.90 25,443.20

20,215.26 21,255.51 20,524.54

21,317.00 22,419.40 22,051.40

2.12 0.40 1.33

20,482.65 21,457.48 20,764.05

– – –

1.25 -0.30 0.77

-0.74 4.96 -0.19

20,215.26 21,255.51 –

25,453.80 26,071.00 25,844.20

20,066.40 22,310.60 20,496.40

– – 20,535.70

25,670.90 26,253.76 25,988.46

0.33 0.07 -0.29

3.48 2.34 2.51

-0.45 -0.94 -1.08

3.09 1.43 1.95

25,024.46 25,904.29 25,370.40

25,915.10 26,524.50 26,146.00

25,221.09 26,167.34 25,572.81

26,013.09 26,762.80 26,291.65

0.77 0.07 -0.11

6.05 2.15 4.15

– – –

-0.26 -1.20 -0.86

26,658.10 26,712.40 26,712.40

– – –

25,072.36 25,835.85 25,425.85

– – –

– – –

25,331.66 26,166.99 25,617.77

– – –

– – –

-0.69 1.40 -0.79

25,138.40 26,149.90 25,647.00

8.09 7.01 8.02

2.23 2.33 2.96

5.96 5.61 6.51

0.55 0.66 1.58

36,037.90 39,339.70 37,087.00

– – – – – – – 1.52

34,197.08 37,494.91 35,369.82

36,762.59 39,860.18 37,613.59

– – – – – – –

34,770.70 38,116.58 35,852.31

– – –

– – – 1.82

1.06 1.11 1.15

– – –

– – – 0.92

0.00 0.00 0.00

34,011.59 37,248.88 34,820.79

– – –

34,011.59 37,248.88 34,820.79

36,762.59 39,860.18 37,613.59

– – –

34,372.87 37,662.68 35,220.86

– – –

2.41 7.31 3.92 3.56

0.39 0.15 0.31 1.16

2.07 6.41 3.57 2.71

0.02 -0.06 0.02 0.19

132,094.00 154,237.00 137,213.00

129,451.35 144,852.83 132,511.81

132,537.10 155,542.00 137,677.00

129,931.15 145,163.65 132,902.16

– – – 5.74

0.35 0.23 0.34 0.66

– – –

0.00 0.00 0.00 -0.11

129,420.62 144,945.35 132,484.84

129,420.62 144,945.35 132,484.84

132,537.10 155,542.00 137,677.00

129,877.75 145,280.40 132,930.21

– – –

Lum et al. [33]

Table 9: Comparison of the results for the VeRoLog Solver Challenge instances Test instance

small all with trailer all without trailer normal medium all with trailer all without trailer normal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal Average

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BKS = ((Avg. Sol. - BKS)/BKS)*100). Note that BKSs are shown in the second, third and fourth column in Table 9. Average evaluation (Avg. eval.) corresponds to the average number of solutions which is investigated through the search. The solutions are underlined when the BKS is improved. In Table 9, the first column represents the name of the benchmark instance and in the second, third as well as the fourth column best known solutions are illustrated. For some instances, such as medium – normal, large1 – normal, large2 – all with trailer, large2 – all without trailer and large2 – normal, BKSs were reported and verified by the organizers. For the other instances, the BKSs are determined by the work of [33, 34, 26] (third column). Since only the results of [34] are available for the final and final random instance sets, we conducted experiments for those instances based on the approach in [26]. However, the results for the final and final random instances are separated to show that they have not been published before. An overall update of the BKSs is given in Section 5.7. Furthermore, we run the code with the settings presented in [26], whereas the neighborhood operators are selected randomly (shown as “Our Random VNS” in Table 9). Table 9 shows that, on average, the Gap BKS of [33] is relatively high with 5.74, followed by the results of [34] with a Gap BKS of 2.71. Then, the studies of [51] and [1] come next with a Gap BKS of 1.52 respectively 0.92. The lowest Gap BKS is, on average, realized by the approaches “Our VNS” and “Our Random VNS”. Our VNS performed well on the datasets since 4 out of 18 BKSs were improved, 11 BKSs were identified and the average gap with respect to the BKSs was -0.11%. This indicates that the solutions are, on average, of high quality. Our VNS visits, on average, more than 1 billion solutions during the running time of 600 seconds. Since a high degree of practical relevance should be achieved, the computational time of 600 seconds is not differentiated for the small-sized, medium and large instances. We also applied “Our Random VNS” where the same neighborhood operators are used as in the VNS [26] such as 2-EX, CSL, INTRA, 2-OPT and INTER. However, the order of the operators is chosen randomly. The statistics in Table 9 demonstrate that the results are worse than “Our VNS” results with a 0.19% gap BKS and 1.16% average gap BKS. Our main remark is that the order of the neighborhood operators in the VNS is important and putting effort in testing the sequences leads to high quality solutions.

Impact of the synchronisation times on the solution quality

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Synchronisation times (denoted as synctime in Figure 7) are the first aspect to be investigated in more detail. This value controls the exchange of the best solution between the 4 CPU cores. Let us e. g. assume a value of 600 seconds which would mean that no information exchange would happen during the optimization run. The lowest average solution respectively the best solution over all synchronisation times is identified for each test instance. Based on this value, the deviation values are calculated. Then, the deviation values are summed up over all instances for each synchronisation time. In Figure 7 the results are presented. Thereby, the results are evaluated on the average (avg.) and the best solutions (Best Sol.). Also, 2 variants are considered: 1) the small instances are integrated in the calculation (Avg.-small and Best Sol.-small) or they are not included (Avg.-Not-small and Best Sol.-Not-small). This distinction is made since large instances are more relevant for the practical applications. It can been seen in Figure 7 that the lowest avg. total sum is achieved for synctime 2 seconds (Avg.-Not-small). However, without the small instances synctime 1 second and 2 seconds are equally good (Avg.-small). Synctime 2 seconds also gives the lowest total sum considering the best solutions (Best Sol.-small). When the small instances are not included, synctime 1 second is slightly better. Based on these tests, it can be verified that the algorithm performs better when synctime 2 seconds is applied. It seems that more information exchanges between the CPU cores are beneficial as opposed to letting them run longer separately. Higher synctimes such as 16 seconds/17 seconds cannot improve the solution quality and are therefore not further tested.

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10.00% 9.00% 8.00% 7.00% 6.00% 5.00% 4.00% 3.00% 2.00%

0.00%

Avg.-Not-small

Avg.-small

Best Sol.-Not-small

Best Sol.-small

Figure 7: Minimize the total sum of deviation values (%)

Impact of the perturbation on the solution quality

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In Table 10 the performance of the algorithm is summarized with respect to the perturbation. The first column reports the removal of one entire tour, the second column removes two tours etc. Tests show that destroying two tours usually provide a better solution quality. This is true for 17 out of 18 instances when the results of removing two tours are compared with eliminating one tour. Also, the deletion of two tours yields better results than the destruction of three respectively four tours. Thus, the removal of two tours is most successful to overcome local optima.

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Table 10: Applying the perturbation by removing entire tours in order to overcome local traps

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small all with trailer all without trailer normal medium all with trailer all without trailer normal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal

5.6

1 tour Avg. Sol.

2 tours Avg. Sol.

3 tours Avg. Sol.

4 tours Avg. Sol.

4,730.92 4,909.92 4,818.30

4,731.02 4,869.05 4,808.32

4,730.72 4,883.33 4,802.23

4,730.70 4,848.21 4,803.36

7,851.25 8,128.63 7,887.64

7,785.26 8,075.92 7,876.85

7,831.21 8,137.71 7,916.79

7,918.89 8,221.42 7,988.08

20,377.96 21,490.45 20,490.91

20,360.38 21,387.95 20,600.80

20,644.74 21,480.69 20,793.47

20,882.63 21,750.62 21,001.61

25,293.50 26,296.97 25,666.04

25,033.79 26,050.31 25,372.88

25,508.00 26,355.57 25,758.44

25,748.79 26,681.77 26,067.10

34,390.82 38,650.89 35,374.00

34,244.68 37,691.03 35,201.07

34,990.34 38,264.65 35,898.63

35,652.84 38,694.21 36,410.94

130,217.73 145,705.88 133,313.30

129,888.99 145,215.62 132,919.32

129,973.96 145,295.77 132,914.85

130,126.55 145,423.59 133,078.78

Impact of the neighborhood operators on the solution quality

As analyzed in Section 5.3 and based on the performance in the VeRoLog Challenge it can be concluded that the VNS is competitive and achieves high quality solutions. However, it is not clear how the different neighborhood operators contribute to the solution quality. Thus, several questions arise such as 1) which neighborhood operator yields the most gain, 2) does a preferable sequence of operators exist that outperforms other sequences and 3) what is the ideal or at least a good number of operators.

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An overview of the investigated sequences Hλ,l are presented in Table 11 whereas λ = 1, . . . , 6 gives the number of used neighborhood operators and l identifies the sequence. Table 11: The investigated sequences of the neighborhood operators Hλ,l within the VNS.

H1,1 H1,2 H1,3 H1,4 H1,5 H1,6 H1,7 H2,1 H2,2 H2,3 H2,4 H2,5 H2,6 H2,7 H2,8 H2,9 H2,10 H2,11 H2,12

Sequence of the neighborhood operators One operator 2EX CSL INTER INTRA 2OPT 3OPT 3EX Two operators 2EX-INTER 2EX-CSL 2EX-2OPT 2EX-3EX 2EX-3OPT 2EX-INTRA INTER-2EX INTER-CSL INTER-2OPT INTER-3EX INTER-3OPT INTER-INTRA

Hλ,l

H3,1 H3,2 H3,3 H3,4 H3,5 H4,1 H4,2 H4,3 H4,4 H5,1 H5,2 H5,3 H5,4 H5,5 H5,6

Sequence of the neighborhood operators

Three operators 2EX-INTER-2OPT 2EX-INTER-3EX 2EX-INTER-3OPT 2EX-INTER-INTRA 2EX-INTER-CSL Four operators 2EX-INTER-2OPT-INTRA 2EX-INTER-2OPT-3OPT 2EX-INTER-2OPT-3EX 2EX-INTER-2OPT-CSL Five operators 2EX-INTER-2OPT-CSL-INTRA 2EX-INTER-2OPT-CSL-3OPT 2EX-INTER-2OPT-CSL-3EX 2EX-INTER-2OPT-INTRA-3OPT 2EX-INTER-2OPT-INTRA-3EX 2EX-INTER-2OPT-INTRA-CSL

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Investigating appropriate combinations of neighborhood operators is a difficult issue. This is due to the fact that e. g. not all combinations can be tested since the number of sequences is high. 7! = 5,040 possible sequences exist for the seven neighborhood operators. Exemplarily, the runtime for testing one sequence for all instances is time consuming with 324,000 seconds (600 seconds per instance * 18 instances * 30 runs). An overview of the experimental setting is presented in Figure 8. Firstly, only one neighborhood operator is used in our VNS, where the 2EX and the INTER operator yields promising results. Thus, both operators are investigated in more detail. In Figure 8 only the investigation of each combination with the 2EX operator is illustrated. Secondly, sequences based on the 2EX followed by the INTER operator are tested. The combination 2EX → INTER → 2OPT is further analyzed. This continues with the 2EX → INTER→ 2OPT→ INTRA and with the 2EX → INTER → 2OPT→ CSL sequence. After that, the investigated combinations cannot obtain an improvement on the objective function. Detailed results are described in the following.

AC

2EX→CSL

3OPT

2EX→INTRA

2EX→3EX

2EX→INTER→3OPT

2OPT

3EX

2EX→INTER

2EX→INTER→INTRA

2EX→INTER→2OPT →INTRA

…

2EX

CSL

INTRA

2EX→3OPT

2EX→INTER→2OPT

2EX→INTER→ 2OPT →3OPT

2EX→INTER→ 2OPT → CSL → INTRA

INTER

2EX→2OPT

2EX→INTER→3EX

2EX→INTER→ 2OPT →CSL

2EX→INTER→ 2OPT → CSL →3EX

…

2EX→INTER→CSL

2EX→INTER→ 2OPT →3EX

2EX→INTER→ 2OPT → CSL → 3OPT

Figure 8: Overview of the experimental setting. The highlighted fields show which sequences should be further investigated.

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H1,5 2OPT Avg. Avg. Gap Sol. BKS

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H1,6 3OPT

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H1,7 3EX

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Test instance Avg. Avg. Gap Sol. BKS

806,834,580 1.12 1,040,154,718 2.22 862,616,359 0.91

Avg. eval.

4,783.40 4,947.25 4,840.99

Avg. Sol.

1,204,296,621 3.19 846,912,920 1.71 969,466,800 2.02

(known so far)

1,783,805,261 1.23 1,053,056,904 1.87 1,496,375,001 2.07

4.37 5.14 5.35

4.01 134,669.56 17,185,840 6.15 154,335.36 3,454,324 5.26 139,759.51 10,178,306 9.45

4,881.59 4,922.33 4,894.38

1,520,180,284 0.44 1,746,747,317 5.11 2,026,158,428 1.32

778,839,859 820,113,126 775,759,772

26,761.16 28,332.24 27,440.01

4,788.98 4,930.17 4,896.91

2,117,492,113 0.07 4,751.26 2,662,711,952 1.25 5,086.78 2,314,301,346 0.35 4,860.86

1,160,159,557 12.98 8,094.23 1,143,211,194 13.78 8,458.95 1,138,955,031 14.41 8,214.05

7.38 9.10 7.22 25,600.06 26,498.32 25,930.19

4,733.97 4,899.93 4,814.45

40,500,371 0.69 1,769,385,789 2.86 1,106,088,611 4.67

1,899,282,722 9.14 8,761.82 1,729,759,168 11.32 9,154.12 1,774,302,737 12.44 8,920.51

753,862,530 792,157,177 792,065,895

1,325,978,665 0.07 4,763.34 1,535,134,000 1.69 4,977.93 1,374,383,093 1.93 5,021.47

1,515,099,687 8.80 8,464.49 1,721,394,077 10.15 8,956.09 1,985,875,235 11.51 8,766.70

19.11 21,707.95 23.70 23,190.29 19.51 22,017.59

4,733.99 4,921.24 4,890.07

1,980,001,448 1.54 8,437.60 2,414,157,962 1.53 8,862.14 2,055,671,573 2.09 8,694.23

961,336,183 861,790,682 923,583,049

4,730.63 4,839.64 4,797.55

12,753,287 10.12 7,874.74 1,284,666,750 10.11 8,168.40 768,374,306 12.84 7,959.61

1,234,242,947 13.91 24,077.49 1,116,824,894 18.73 26,292.17 1,155,344,241 15.40 24,541.48

1,176,976,100 2.96 8,540.39 1,300,030,525 2.71 8,859.15 1,191,775,641 3.39 8,798.06

14.08 23,026.31 17.52 25,237.01 14.61 23,697.52

8.62 15.47 8.47

7,984.86 8,263.24 8,061.22

611,386,940 619,616,667 861,056,064

860,925,750 903,133,040 878,712,222

7,755.43 8,045.47 7,796.68

2,012,591,994 2.83 23,060.83 2,455,470,030 3.36 24,980.49 2,087,606,081 3.86 23,535.82

11.29 27,306.06 12.50 30,194.61 11.22 27,819.73

4,247,878 14.54 20,786.80 1,123,293,313 18.78 21,969.75 460,973,060 15.74 21,327.68

857,015,895 789,463,441 832,924,157

1,060,082,766 4.36 23,153.86 1,182,992,832 3.52 25,248.28 1,082,018,397 3.40 23,768.43

27,977.52 29,417.73 28,523.45

21,096.84 22,003.27 21,234.12

1,023,190,906 6.46 932,515,762 8.35 932,143,787 6.99

20,215.26 21,255.51 20,535.70

6.84 8.71 7.09

2,243,560,102 1.84 26,857.41 2,486,009,451 1.33 28,427.60 2,290,167,561 1.10 27,466.11

458,749,763 390,489,938 562,731,311

4,138,139 6.84 1,001,392,799 9.11 311,393,644 7.65

8.92 10.94 9.15

1,150,825,584 1.40 26,858.74 1,195,625,025 1.66 28,533.26 1,163,660,519 1.04 27,609.02

773,682,079 779,867,125 786,005,562

25,490.14 26,583.73 25,914.06

22.17 37,046.35 23.18 41,323.76 22.59 38,006.39

25,138.40 26,149.90 25,647.00

943,571,224 881,697,910 903,401,617

1,930,058,986 3.64 39,066.93 2,388,483,605 3.91 43,662.21 2,008,283,981 4.69 40,497.48

1,247,616,046 14.08 41,552.76 1,220,108,496 16.92 45,883.06 1,165,174,295 15.79 42,687.65

4,415,097 14.78 35,250.98 1,193,567,351 17.90 38,704.14 431,898,032 16.19 36,455.19

4.13 134,605.68 62,257,618 6.46 153,863.09 18,774,548 5.18 139,449.49 46,252,693 9.46

4.81 6.55 5.62 6.74

14.86 38,799.71 17.22 43,550.98 16.30 40,319.56

1,133,487,502 3.19 39,038.77 1,224,004,700 3.96 43,915.69 1,157,121,650 4.78 40,458.79

1.03 134,966.92 3,714,717 4.29 131,958.28 627,407,778 1.96 134,761.81 133,338,293 1.84 153,668.19 1,836,654,932 6.02 146,924.92 1,623,617,705 1.37 154,315.50 114,238,760 1.45 139,329.49 283,376,341 5.17 134,743.65 633,130,559 1.70 139,350.39 145,687,278 2.47 9.91 2.13

4.06 135,645.59 195,145,876 6.48 154,443.69 508,828,255 5.49 139,929.28 226,124,468 12.74

545,361,123 507,405,222 738,485,791

35,097.78 38,724.85 36,485.78

129,420.62 130,751.22 602,395,697 144,945.35 147,619.28 920,224,272 132,484.84 134,407.72 625,049,451

34,011.59 37,248.88 34,820.79

Table 12: Average solutions and average number of evaluations for each neighborhood operator

small all with trailer all without trailer normal medium all with trailer all without trailer normal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal Average

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BKS Avg. Sol.

1,600,518,461 1,968,680,096 1,751,632,586

Avg. eval.

H2,1 2EX-INTER

0.37 0.56 0.76

0.02 0.54 0.20

Avg. Gap BKS

21,124.80 22,051.64 21,255.73

7,980.48 8,266.39 8,040.16

4,733.99 4,931.70 4,813.26

Avg. Sol.

1,219,998,535 1,320,790,024 1,237,053,285

1,123,869,306 1,397,183,784 1,154,273,713

1,245,932,244 1,367,743,249 1,223,817,243

1,392,613,891 1,744,804,436 1,548,720,655

Avg. eval.

3.03 4.77 4.61

1.34 1.99 0.74

4.50 3.75 3.51

2.90 2.75 3.12

0.07 1.90 0.33

Avg. Gap BKS

130,700.21 147,638.33 134,613.69

35,202.85 38,701.83 36,383.66

25,437.65 26,520.06 25,901.63

21,144.19 21,900.93 21,186.43

7,990.28 8,269.14 8,017.53

4,754.19 4,893.62 4,859.29

Avg. Sol.

651,156,482 966,584,472 675,663,004

1,288,901,591 1,379,807,640 1,313,399,294

1,288,629,498 1,330,329,216 1,304,786,412

1,249,866,480 1,327,755,051 1,272,559,711

1,362,966,817 1,449,290,142 1,376,170,070

1,536,874,229 1,653,144,123 1,569,108,936

Avg. eval.

0.99 1.86 1.61 2.35

3.50 3.90 4.49

1.19 1.42 0.99

4.60 3.04 3.17

3.03 2.78 2.83

0.50 1.12 1.29

Avg. Gap BKS

134,542.62 152,957.30 139,168.15

36,289.37 40,369.97 37,665.80

26,519.47 28,719.44 27,129.23

21,499.52 22,962.55 21,678.92

8,039.37 8,378.39 8,101.79

4,757.77 4,912.76 4,818.22

Avg. Sol.

213,752,589 548,145,052 250,967,767

862,873,206 860,210,569 854,474,940

953,564,911 967,538,610 961,396,631

823,992,461 857,505,029 842,686,139

867,035,554 920,683,947 861,149,850

914,292,224 1,073,910,074 1,017,772,732

Avg. eval.

3.96 5.53 5.04 5.17

6.70 8.38 8.17

5.49 9.83 5.78

6.35 8.03 5.57

3.66 4.14 3.91

0.57 1.51 0.43

Avg. Gap BKS

130,694.63 147,562.91 134,423.65

36,080.74 39,367.03 36,993.55

25,562.27 26,725.26 26,102.31

21,229.80 22,276.95 21,365.20

8,046.71 8,311.65 8,083.60

4,750.61 4,934.90 4,858.17

Avg. Sol.

576,648,404 883,202,545 603,035,841

1,062,136,618 1,097,034,434 1,068,908,381

1,082,328,640 1,109,627,641 1,090,841,154

1,057,712,066 1,074,727,678 1,060,523,984

1,125,006,687 1,163,617,468 1,130,267,813

1,277,946,804 1,425,265,844 1,323,009,701

Avg. eval.

0.98 1.81 1.46 3.12

6.08 5.69 6.24

1.69 2.20 1.78

5.02 4.81 4.04

3.76 3.31 3.68

0.42 1.97 1.26

Avg. Gap BKS

130,879.79 147,486.51 134,566.43

35,213.59 38,729.66 36,550.05

25,477.00 26,525.04 25,965.91

20,954.67 22,053.17 21,391.08

7,997.25 8,339.53 8,037.01

4,764.76 4,859.65 4,828.19

Avg. Sol.

655,397,734 948,571,607 666,504,949

1,153,591,282 1,269,221,802 1,184,540,681

1,166,761,123 1,228,698,602 1,187,672,682

1,108,150,112 1,278,887,248 1,136,915,299

1,241,657,531 1,376,617,489 1,260,161,943

1,485,206,762 1,662,828,363 1,553,785,524

Avg. eval.

1.13 1.75 1.57 2.45

3.53 3.98 4.97

1.35 1.43 1.24

3.66 3.75 4.17

3.12 3.65 3.08

0.72 0.41 0.64

Avg. Gap BKS

CR IP T

(known so far)

4,731.58 4,865.65 4,807.30 1,418,305,324 1,674,591,057 1,447,921,071 0.73 0.38 0.42

25,475.20 26,669.44 25,836.98

1,197,260,961 1,397,440,334 1,236,670,690

1.05 1.33 1.44 2.40

AN US

4,730.63 4,839.64 4,797.55 7,784.37 8,090.86 7,855.74 1,465,862,936 1,766,234,932 1,499,375,882 -0.34 -0.24 -0.63

35,043.54 39,026.64 36,426.54 634,766,293 1,556,026,826 804,885,428

Avg. Sol.

2,177,995,743 3,042,779,643 2,520,410,801

Avg. eval.

1.70 1.47 1.67

0.07 0.62 0.68

Avg. Gap BKS

20,768.43 21,874.47 21,180.59

7,875.10 8,176.64 7,959.31

4,733.09 4,934.85 4,808.57

Avg. Sol.

2,295,289,749 2,523,870,171 2,333,927,875

2,098,542,568 2,486,629,096 2,157,688,467

2,169,207,472 2,451,276,176 2,211,781,593

2,419,189,631 2,693,223,072 2,528,864,376

Avg. eval.

3.16 3.35 3.66

1.23 1.05 1.22

2.74 2.91 3.14

1.54 1.63 2.09

0.05 1.97 0.23

Avg. Gap BKS

131,836.35 149,572.18 135,453.60

35,809.48 40,163.44 36,979.01

26,459.64 27,931.61 26,640.61

21,161.09 22,284.37 21,320.96

7,949.29 8,274.49 8,026.54

4,735.43 4,911.77 4,811.85

Avg. Sol.

180,155,615 606,855,171 203,886,133

841,342,435 898,532,145 828,684,535

937,365,366 984,300,252 941,086,644

804,781,680 880,415,380 829,964,614

890,406,135 927,695,002 862,731,990

873,948,414 1,058,087,638 961,604,984

Avg. eval.

1.87 3.19 2.24 3.67

5.29 7.82 6.20

5.26 6.81 3.87

4.68 4.84 3.82

2.50 2.85 2.95

0.10 1.49 0.30

Avg. Gap BKS

131,734.41 146,880.99 134,764.84

35,778.46 39,461.52 37,096.52

25,878.94 26,688.44 26,357.11

21,032.72 22,138.90 21,438.16

7,912.99 8,268.54 7,976.76

4,752.57 4,918.00 4,848.88

Avg. Sol.

660,531,603 1,637,664,197 663,797,866

1,307,040,766 1,495,228,916 1,338,503,502

1,520,593,205 1,687,576,625 1,559,644,498

1,365,830,527 1,585,622,636 1,385,295,523

1,326,467,155 1,459,224,967 1,336,668,429

1,696,447,822 2,035,250,247 1,852,290,403

Avg. eval.

1.79 1.34 1.72 2.95

5.19 5.94 6.54

2.95 2.06 2.77

4.04 4.16 4.39

2.03 2.77 2.31

0.46 1.62 1.07

Avg. Gap BKS

131,631.03 146,909.65 134,743.10

35,264.95 38,406.84 36,030.16

25,612.21 26,388.16 25,918.80

20,826.17 21,829.04 21,212.68

7,857.27 8,182.51 7,929.84

4,732.84 4,921.58 4,826.98

Avg. Sol.

724,275,780 1,678,739,727 729,794,957

2,055,636,618 2,569,824,894 2,125,273,011

2,341,245,493 2,637,941,291 2,384,267,674

2,121,670,455 2,642,572,168 2,211,753,595

2,281,871,653 2,825,402,360 2,370,841,796

2,698,231,375 3,374,105,942 3,005,284,023

Avg. eval.

1.71 1.36 1.70 1.94

3.69 3.11 3.47

1.88 0.91 1.06

3.02 2.70 3.30

1.31 1.70 1.71

0.05 1.69 0.61

Avg. Gap BKS

M

H2,7 INTER-2EX Avg. Gap BKS

4,733.97 4,869.79 4,830.21

2,076,891,016 2,545,502,765 2,173,472,630

3.09 2.82 3.37

25,447.12 26,424.82 25,960.91

2,010,480,944 2,416,483,447 2,070,259,304

1.74 1.31 1.68 1.93

ED

Avg. eval.

0.01 1.53 0.33

7,887.05 8,164.07 7,927.00

2,068,045,094 2,606,451,975 2,180,716,175

1.77 1.35 1.09

35,086.30 38,498.56 36,094.27

665,984,586 1,572,710,717 667,790,271

PT

BKS Avg. Sol.

1,584,768,408 1,996,334,645 1,717,298,580

0.45 0.69 1.10

20,839.04 21,853.87 21,227.77

2,307,460,662 2,634,138,547 2,356,296,325

3.83 3.71 3.69

131,669.02 146,841.99 134,708.98

H2,6 2EX-INTRA

7,755.43 8,045.47 7,796.68 20,362.75 21,335.63 20,622.00 1,613,978,856 1,741,305,790 1,644,941,317 0.66 1.18 1.48 130,783.85 146,868.12 134,394.76

(known so far)

4,731.02 4,913.61 4,813.25 1,401,893,891 1,632,457,363 1,438,348,285

0.85 0.66 0.69

25,582.65 26,503.83 25,927.18

1,978,649,214 2,562,030,227 2,062,524,268

1.85 0.62 1.57 1.94

H2,5 2EX-3OPT

20,215.26 21,255.51 20,535.70 25,052.90 26,086.04 25,485.91 1,443,112,552 1,711,663,864 1,498,376,341 0.43 0.63 0.43 0.42

H2,4 2EX-3EX

25,138.40 26,149.90 25,647.00 34,235.47 37,689.61 35,337.54 594,578,506 1,340,770,256 625,840,445

H2,3 2EX-2OPT

34,011.59 37,248.88 34,820.79 129,982.83 145,860.19 133,053.88

H2,2 2EX-CSL

129,420.62 144,945.35 132,484.84

4,730.63 4,839.64 4,797.55 7,790.00 8,101.07 7,882.56

1,430,879,205 1,694,277,965 1,465,101,506

0.11 -0.09 -0.33

35,314.21 38,631.03 36,104.04

656,931,808 2,162,490,114 935,316,298

H2,12 INTER-INTRA

7,755.43 8,045.47 7,796.68 20,386.14 21,396.56 20,676.57

1,572,600,086 1,691,154,746 1,604,129,876

2.02 1.32 2.17

131,820.59 145,848.39 134,559.17

H2,11 INTER-3OPT

20,215.26 21,255.51 20,535.70 25,167.09 26,125.20 25,562.84

1,428,683,607 1,651,913,588 1,470,139,676

0.41 0.64 0.30 0.71

H2,10 INTER-3EX

25,138.40 26,149.90 25,647.00

34,699.97 37,742.02 35,577.21

631,596,500 1,309,365,815 652,605,896

H2,9 INTER-2OPT

34,011.59 37,248.88 34,820.79

129,946.58 145,870.52 132,876.29

H2,8 INTER-CSL

129,420.62 144,945.35 132,484.84

CE

Test instance

Table 13: Combination with 2 neighborhood operators on the basis of the 2EX neighborhood operator

small all with trailer all without trailer normal medium all with trailer all without trailer normal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal Average

Test instance

Table 14: Combination with 2 neighborhood operators on the basis of the INTER neighborhood operator

small all with trailer all without trailer normal medium all with trailer all without trailer normal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal Average

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ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

22

AC

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AN US

CR IP T

In order to observe the contribution of each operator, the algorithm solely uses the investigated neighborhood operator. This observation seems to be important since the work of [37] showed that a VNS with only one neighborhood operator outperforms a complex hybrid metaheuristic. Tables 12-18 have a similar structure. The first column gives the name of the instance, the second shows the BKS and then the average solution, the avg. evaluations and the average Gap BKS is presented for the proposed sequence. Table 12 shows the superiority of the 2EX and INTER neighborhood operator with a Gap BKS of 2.13 and 2.47. Furthermore, the third inter-tour operator, 3EX, achieves good results. For the investigated problem, the results suggest that the inter-tour operators are essential for the solution method. Analyzing the results of 2EX and INTER with the solutions of [33] and [34] in Table 9, it can be shown that the avg. solutions achieve better results than the avg. solutions of [34] for the medium set, the large2 set, the large1-all with trailer instance, the final set and the final random set. Moreover, our avg. solutions are better than the best solutions of [34] for the final random set, the final set, the large2 set and the medium-all without trailer instance. When comparing our avg. results with the best solutions of [33], one can notice that our algorithm outperforms the solutions of [33] for all instances (small set, medium set, large1 and large2 set). These findings show that the 2EX and INTER are competitive by themselves. However, they are not capable of finding the BKSs when they run alone. Based on the findings in Table 12, the 2EX and the INTER operator are investigated in more detail. In Table 13, it is illustrated that the sequence H2,1 clearly outperforms the other sequences for the small, medium, large1, large2, final as well as final random set and has the lowest avg. Gap BKS with 0.42. It can also be observed that the sequence matters since H2,7 performs worse with the exception of the final random instance set (see Table 14). This first analysis strongly suggests the further investigation of the H2,1 sequence. The results based on the sequence H2,1 plus an additional operator are depicted in Table 15. It is shown that the sequence H3,1 achieves the best avg. results for 13 instances. However, for both the final/ final random all without trailer instances and the small instance set the sequence H3,4 respectively H3,3 achieve better results. Sequence H3,1 is capable of improving the BKS for the large2 set. This is also true for the sequence H3,4 and H3,5 . The small all with trailer and the large2 normal instance could additionally be improved by the sequence H3,3 . Comparing these findings with the results of the VNS [26] in Table 9, it can be reported that the sequence H3,3 outperforms the avg. solutions for the small normal and the small all with trailer instances. Furthermore, the BKS is surpassed by H3,3 for the small all with trailer instance. It appears as if for the smaller instances with up to 50 customers the sequence H3,3 with the combination of a more sophisticated operator such as a 3OPT obtains better results than e. g. the sequence with 5 operators (excluding the 3OPT) used in the VeRoLog Challenge (see Table 9) or H5,2 (including 3OPT) in Table 17. Regarding the average number of evaluations, the sequence H3,3 investigates less solutions than the sequence H3,1 and H3,4 . Still, the number of average evaluations is high with around 1 billion. Besides, the sequence H3,2 has a lower avg. number of evaluations which is not resulting in improved solutions. Overall, the sequence H3,1 is competitive with respect to the solution quality and is therefore analyzed in more detail. The results based on the sequences H3,1 are summarized in Table 16. Thereby, the Gap (l, H3,1 ) calculates the gap between the investigated average solution (H4,l ) and the average solution of the best sequence with only three operators H3,1 (Gap (l, H3,1 ) = ((Avg. Sol.(H4,l ) - Avg. Sol.(H3,1 ))/Avg. Sol.(H3,1 ))*100) with l = 1, . . . , 4. These calculations are used to analyze the effect on the solution quality when a neighborhood operator is added to the best previously investigated sequence. One can observe, in Table 16, that the order H4,1 , H4,2 and H4,4 could improve the BKS in case of the large2 instance set. Moreover, the sequence H4,4 is capable of improving the BKS for the large1 all without trailer instance with an average gap BKS of -0.10% and for the final normal instance with an average gap BKS of -0.49%. Integrating an extra operator is particularly advantageous for the order H4,1

BKS Avg. Sol

1,691,004,307 1,967,835,238 1,790,609,423

Avg. eval

H3,1 2EX-INTER-2OPT

0.51 0.72 1.08

0.01 0.93 0.16

Avg. Gap BKS

20,986.68 22,077.51 21,315.21

7,872.51 8,256.62 7,977.12

4,734.24 4,914.06 4,811.77

Avg. Sol

872,220,494 910,673,869 863,680,994

763,630,693 826,119,803 768,870,146

828,875,670 878,765,763 810,969,517

844,517,898 1,115,475,375 906,623,935

Avg. eval

4.79 6.39 4.95

3.30 5.33 3.23

3.82 3.87 3.80

1.51 2.62 2.31

0.08 1.54 0.30

Avg. Gap BKS

129,948.81 145,848.78 132,936.00

34,896.91 37,939.37 35,599.57

25,258.87 26,199.81 25,552.23

20,532.48 21,477.97 20,818.30

7,812.48 8,127.38 7,918.75

4,730.00 4,869.23 4,800.43

Avg. Sol

567,546,105 1,491,870,006 584,218,581

1,110,612,306 1,239,582,822 1,126,817,591

1,231,777,242 1,335,561,038 1,257,977,612

1,134,421,632 1,276,288,800 1,151,331,044

1,150,286,869 1,266,162,484 1,156,542,875

1,373,816,199 1,662,466,382 1,478,188,768

Avg. eval

0.41 0.62 0.34 0.91

2.60 1.85 2.24

0.48 0.19 -0.37

1.57 1.05 1.38

0.74 1.02 1.57

-0.01 0.61 0.06

Avg. Gap BKS

129,985.03 145,831.96 132,949.33

34,316.03 37,626.56 35,343.55

25,131.32 26,061.96 25,523.91

20,439.57 21,356.01 20,635.18

7,796.54 8,110.91 7,885.28

4,730.92 4,874.36 4,804.54

Avg. Sol

679,582,309 1,680,136,380 701,918,931

1,454,651,764 1,737,498,577 1,519,588,839

1,600,037,771 1,762,543,150 1,647,105,623

1,485,952,281 1,803,473,201 1,531,867,403

1,480,617,756 1,726,865,185 1,513,802,590

1,747,105,746 2,112,184,071 1,924,967,368

Avg. eval

0.44 0.61 0.35 0.52

0.90 1.01 1.50

-0.03 -0.34 -0.48

1.11 0.47 0.48

0.53 0.81 1.14

0.01 0.72 0.15

Avg. Gap BKS

129,941.58 145,275.74 133,081.57

34,362.47 37,797.97 35,354.45

25,075.30 26,131.08 25,541.97

20,382.25 21,408.35 20,644.49

7,793.55 8,095.73 7,870.52

4,731.60 4,868.88 4,807.96

Avg. Sol

565,444,250.43 1,568,060,878.80 723,073,359.97

1,300,889,761.93 1,568,230,937.90 1,358,505,353.90

1,457,024,145.77 1,597,927,794.93 1,484,745,495.97

1,323,736,712.90 1,650,606,237.90 1,354,898,199.17

1,273,142,993.90 1,514,770,888.30 1,299,260,616.07

1,445,991,343.30 1,859,854,498.90 1,591,617,825.70

Avg. eval

0.40 0.23 0.45 0.52

1.03 1.47 1.53

-0.25 -0.07 -0.41

0.83 0.72 0.53

0.49 0.62 0.95

0.02 0.60 0.22

Avg. Gap BKS

CR IP T

(known so far)

4,731.02 4,884.80 4,805.44 1,487,884,865 1,664,782,383 1,516,982,479 0.87 0.42 0.21

25,968.81 27,543.98 26,475.49

793,838,334 802,775,538 771,763,491

1.40 2.74 2.08 3.00

AN US

4,730.63 4,839.64 4,797.55 7,794.96 8,103.68 7,880.50 1,494,545,012 1,760,392,042 1,540,782,489 -0.43 -0.67 -0.74

35,641.47 39,628.31 36,543.44 189,217,037 735,520,359 225,149,651

H3,5 2EX-INTER-CSL

7,755.43 8,045.47 7,796.68 20,391.88 21,344.14 20,577.83 1,601,963,762 1,738,952,280 1,639,055,617 0.69 1.28 1.31 131,226.55 148,913.07 135,237.16

H3,4 2EX-INTER-INTRA

20,215.26 21,255.51 20,535.70 25,030.07 25,973.88 25,457.03 1,470,037,860 1,692,997,475 1,510,396,316 0.36 0.63 0.31 0.42

H3,3 2EX-INTER-3OPT

25,138.40 26,149.90 25,647.00 34,246.72 37,723.89 35,275.90 636,781,545 1,626,238,734 651,527,426

H3,2 2EX-INTER-3EX

34,011.59 37,248.88 34,820.79 129,884.79 145,860.86 132,894.68

H4,4 2EX-INTER-2OPT-CSL

129,420.62 144,945.35 132,484.84

M

Avg. Sol

129,955.32 145,287.74 132,646.80

34,247.14 37,369.04 34,649.78

25,017.63 26,119.91 25,311.34

649,221,317.00 1,748,351,758.00 815,844,837.00

1,517,945,913.00 1,728,901,602.00 1,531,147,368.00

1,655,071,432.00 1,815,534,005.00 1,679,234,946.00

0.41 0.24 0.12 0.19

0.69 0.32 -0.49

-0.48 -0.11 -1.31

0.05 -0.39 -0.19 -0.23

0.00 -0.94 -1.77

-0.05 0.56 -0.57

PT

CE

ED

Test instance

Table 15: Average solutions, average number of evaluations and Avg. Gap BKS for the sequence based on 2EX-INTER plus an additional operator

small all with trailer all without trailer normal medium all with trailer all without trailer normal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal Average

H4,1 2EX-INTER-2OPT-INTRA

Gap (2, H3,1 )

H4,3 2EX-INTER-2OPT-3EX

BKS

Avg. Gap BKS

Gap (4, H3,1 )

Avg. eval

Avg. Gap BKS

Avg. Sol

Avg. eval

Gap (1, H3,1 )

Avg. Sol

Avg. Gap BKS

Gap (3, H3,1 )

Avg. eval

4,741.19 4,903.81 4,806.47

Avg. Gap BKS

Avg. Sol

0.00 -0.80 -0.04

Avg. eval

(known so far)

0.01 0.13 0.13

0.00 -0.92 0.06

1,645,434,099.00 1,905,599,161.00 1,741,254,092.00

0.01 0.00 0.22

4,731.02 4,845.71 4,803.76

1,713,027,233.00 2,102,221,726.00 1,814,372,098.00

0.00 -0.17 -0.07

894,431,404.00 959,396,344.00 888,070,226.00

4,731.02 4,839.64 4,808.32

0.01 0.76 0.10

7,844.43 8,201.91 7,952.33

0.22 0.39 0.02

1,871,420,677.00 2,179,628,453.00 1,989,361,513.00

0.30 0.21 0.10

0.22 1.33 0.19

4,731.02 4,876.45 4,802.12

0.81 0.93 1.17

937,740,075.00 1,166,011,051.00 1,009,343,244.00

4,730.63 4,839.64 4,797.55

1,330,633,945.00 1,444,262,246.00 1,336,966,047.00

-0.39 0.31 0.59

7,818.58 8,120.66 7,888.28

849,614,398.00 909,799,411.00 866,138,942.00

3.39 3.76 3.98

0.12 1.04 1.68

0.03 -0.26 -0.15

20,953.64 22,078.04 21,137.18

962,457,389.00 984,497,582.00 959,104,325.00

4.11 5.08 5.34

0.86 2.07 1.49 2.22

1,520,243,090.00 1,747,796,110.00 1,557,666,122.00

0.54 0.46 0.92

0.64 0.53 0.88

26,037.77 27,274.29 26,231.14

868,867,564.00 906,387,033.00 860,531,540.00

1.23 2.71 1.81 2.65

7,764.55 8,128.90 7,927.37

1,568,115,329.00 1,782,157,750.00 1,602,141,811.00

1.52 0.95 1.09

0.37 0.57 0.34

35,408.08 39,141.73 36,680.26

201,680,860.00 651,489,771.00 246,369,050.00

0.63 1.21 0.91

7,796.94 8,082.82 7,868.59

1,289,775,073.00 1,453,986,949.00 1,307,798,468.00

-0.07 -0.11 -0.40

1.41 0.59 1.39

131,007.84 148,880.22 134,880.38

1.15 1.94 2.00

7,755.43 8,045.47 7,796.68

20,522.25 21,457.99 20,759.04

1,404,390,559.00 1,501,018,015.00 1,425,242,885.00

2.11 1.87 2.72

0.05 -0.03 0.05 0.36

0.05 -0.51 -0.07

-0.17 -0.23 0.10

25,121.61 26,121.78 25,543.93

1,275,903,616.00 1,415,095,358.00 1,293,114,044.00

0.41 0.60 0.36 0.79

0.92 -0.10 0.14

0.71 0.18 0.30

-0.18 0.06 -0.05

34,730.66 37,945.83 35,767.37

625,828,709.00 1,333,322,804.00 646,120,168.00

1,532,670,195.00 1,807,955,131.00 1,574,675,229.00

1,566,408,904.00 1,837,700,420.00 1,618,303,945.00

-0.61 -0.61 -0.79

0.30 -0.34 -0.14

129,951.28 145,810.83 132,964.12

20,401.61 21,234.74 20,563.91

20,357.80 21,294.75 20,597.40

1,658,212,703.00 1,804,894,745.00 1,702,776,945.00

0.99 0.93 1.16

-0.03 0.01 0.05 -0.07

2.75 3.44 2.72

20,215.26 21,255.51 20,535.70 24,986.01 25,989.71 25,444.79

1,537,004,842.00 1,772,024,260.00 1,583,047,157.00

0.33 0.64 0.36 0.35

3.65 3.87 2.93

25,138.40 26,149.90 25,647.00

34,347.83 37,595.08 35,225.57

702,615,135.00 1,414,286,979.00 718,687,210.00

4.03 5.01 3.04

34,011.59 37,248.88 34,820.79

129,844.31 145,871.50 132,964.60

3.58 4.30 2.28

129,420.62 144,945.35 132,484.84

H4,2 2EX-INTER-2OPT-3OPT

Table 16: Average solutions and average number of evaluations on the basis of 2EX-INTER-2OPT Test instance

small all with trailer all without trailer normal medium all with trailer all without trailer nolmal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal Average

AC

23

ACCEPTED MANUSCRIPT

4,730.63 4,839.64 4,797.55

(known so far)

BKS

7,803.12 8,084.88 7,856.00

4,731.02 4,855.03 4,805.06

Avg. Sol

1,566,121,346.00 1,903,640,553.00 1,618,143,134.00

1,577,888,139.00 1,825,600,611.00 1,623,586,449.00

1,894,222,184.00 2,278,042,384.00 2,025,345,379.00

Avg. eval

-0.47 -0.61 -0.65

0.76 0.47 0.36

0.61 0.49 0.76

0.01 0.32 0.16

Avg. Gap BKS

0.69 1.13 1.75

0.01 -0.50 0.67

-0.16 0.57 0.22

0.50 -0.54 -0.90

0.00 0.32 -0.07

Gap (1, H4,4 )

H5,1 2EX-INTER-2OPT-CSL-INTRA

0.02 -0.45 0.03 0.05

0.40 0.52 0.09

0.13 0.00 0.14

0.06 0.29 0.06

0.08 0.03 -0.16

0.00 -0.44 0.06

Gap (1, H4,1 )

129,808.23 145,324.28 132,989.13

34,767.07 38,041.92 35,421.08

25,122.98 26,112.91 25,578.09

20,511.38 21,410.11 20,644.99

7,815.27 8,108.65 7,885.82

4,731.02 4,894.68 4,810.97

Avg. Sol

621,757,244.00 1,693,668,519.00 774,585,147.00

1,270,433,333.00 1,425,814,824.00 1,271,791,526.00

1,395,063,487.00 1,524,050,059.00 1,413,262,266.00

1,286,050,098.00 1,495,359,331.00 1,302,822,202.00

1,319,026,224.00 1,463,757,140.00 1,335,591,521.00

1,617,051,514.00 1,985,910,230.00 1,739,859,410.00

Avg. eval

0.30 0.26 0.38 0.74

2.22 2.13 1.72

-0.06 -0.14 -0.27

1.46 0.73 0.53

0.77 0.79 1.14

0.01 1.14 0.28

Avg. Gap BKS

-0.11 0.03 0.26 0.56

1.52 1.80 2.23

0.42 -0.03 1.05

0.54 0.83 0.39

0.65 -0.25 -0.52

0.00 1.14 0.06

Gap (2, H4,4 )

-0.03 -0.38 0.02 0.39

1.22 1.19 0.56

0.55 0.47 0.52

0.75 0.54 0.23

0.24 0.32 0.22

0.00 0.37 0.18

Gap (2, H4,1 )

131,034.25 147,880.24 134,698.35

35,501.87 39,088.47 36,448.68

26,000.93 27,274.05 26,428.63

20,953.40 21,935.78 21,189.21

7,855.85 8,215.35 7,951.95

4,743.01 4,902.30 4,803.01

Avg. Sol

203,114,782.00 702,504,006.00 242,396,108.00

860,022,050.00 959,145,272.00 836,301,558.00

957,352,190.00 988,788,082.00 948,378,641.00

843,535,264.00 980,116,108.00 858,663,017.00

905,630,456.00 1,007,372,452.00 891,029,861.00

931,967,450.00 1,220,539,418.00 1,029,912,668.00

Avg. eval

1.25 2.02 1.67 2.60

4.38 4.94 4.68

3.43 4.30 3.05

3.65 3.20 3.18

1.29 2.11 1.99

0.26 1.29 0.11

Avg. Gap BKS

0.83 1.78 1.55 2.41

3.66 4.60 5.19

3.93 4.42 4.41

2.70 3.30 3.04

1.18 1.06 0.31

0.25 1.29 -0.11

Gap (3, H4,4 )

0.92 1.38 1.30 2.24

3.36 3.97 3.47

4.06 4.94 3.87

2.93 3.01 2.87

0.76 1.64 1.06

0.25 0.53 0.02

Gap (3, H4,1 )

CR IP T

7,755.43 8,045.47 7,796.68 20,369.83 21,356.22 20,609.04 1,666,716,501.00 1,833,796,453.00 1,708,751,388.00 1.39 1.45 1.25 -0.07 -0.05 0.27 0.21

AN US

20,215.26 21,255.51 20,535.70 25,019.28 25,989.90 25,480.50 1,531,405,075.00 1,820,255,712.00 1,581,036,663.00 0.35 0.19 0.39 0.40

M

25,138.40 26,149.90 25,647.00 34,484.60 37,790.48 35,256.98 707,984,889.00 1,774,961,457.00 852,718,719.00

H5,3 2EX-INTER-2OPT-CSL-3EX

34,011.59 37,248.88 34,820.79 129,867.97 145,219.59 133,000.19

H5,2 2EX-INTER-2OPT-CSL-3OPT

129,420.62 144,945.35 132,484.84

BKS Avg. Sol

1,730,987,869.00 2,058,384,850.00 1,845,938,690.00

Avg. eval

0.46 0.78 1.62

0.01 0.60 0.14

Avg. Gap BKS

0.53 0.87 1.05

0.35 -0.25 -0.06

0.00 0.60 -0.08

Gap (1, H4,4 )

H5,4 2EX-INTER-2OPT-INTRA-3OPT

0.03 -0.37 -0.23

1.45 0.77 1.19

0.46 0.78 1.62

0.01 0.60 0.14

Gap (1, H4,1 )

35,337.42 39,285.18 36,476.41

25,879.41 27,458.88 26,314.36

20,901.99 22,094.07 21,133.64

7,872.38 8,205.30 7,955.71

4,731.54 4,879.38 4,804.84

Avg. Sol

202,470,821.00 644,853,511.00 242,059,176.00

843,861,583.00 914,867,138.00 835,148,284.00

950,258,304.00 988,373,106.00 948,483,926.00

838,573,950.00 927,782,441.00 825,580,722.00

902,163,755.00 963,023,022.00 875,179,518.00

941,730,420.00 1,142,196,886.00 1,024,844,242.00

Avg. eval

0.94 2.33 1.63 1.86

2.04 3.57 2.13

2.92 5.39 2.84

1.91 3.15 1.70

1.04 1.20 0.42

0.01 0.22 0.01

Avg. Gap BKS

0.88 2.72 1.85 2.46

3.18 5.13 5.27

3.44 5.13 3.96

2.45 4.05 2.77

1.39 0.94 0.36

0.01 0.82 -0.07

0.97 2.31 1.61 2.29

2.88 4.50 3.55

3.58 5.65 3.42

2.67 3.75 2.60

0.97 1.52 1.11

0.01 0.06 0.06

129,897.61 145,295.28 132,912.03

34,324.99 37,744.37 35,201.01

25,009.34 26,107.86 25,428.93

20,377.01 21,344.37 20,577.18

7,785.91 8,092.76 7,851.91

4,731.02 4,906.72 4,804.54

Avg. Sol

708,662,497.00 1,780,877,391.00 853,516,715.00

1,535,724,149.00 1,810,616,318.00 1,590,256,805.00

1,671,431,187.00 1,861,340,707.00 1,712,430,044.00

1,568,815,260.00 1,910,531,402.00 1,622,284,192.00

1,582,140,597.00 1,842,328,950.00 1,620,450,975.00

PT

1,878,243,454.00 2,295,444,855.00 2,012,028,843.00

Avg. eval

0.37 0.24 0.32 0.41

0.92 1.33 1.09

-0.51 -0.16 -0.85

0.80 0.42 0.20

0.39 0.59 0.71

0.01 1.39 0.15

Avg. Gap BKS

-0.04 0.01 0.20 0.22

0.23 1.00 1.59

-0.03 -0.05 0.46

-0.12 0.52 0.06

0.28 -0.44 -0.95

0.00 1.39 -0.08

Gap (3, H4,4 )

0.09 0.45 -0.06

0.09 0.23 -0.10

-0.14 0.12 -0.21

0.00 0.62 0.05

Gap (3, H4,1 )

ED

(known so far)

4,731.02 4,868.80 4,804.25

1,344,787,965.00 1,499,361,146.00 1,368,000,485.00

1.45 0.77 1.19

0.51 -0.25 1.09

1.82 1.84 2.57

131,104.20 149,241.46 135,100.02

H5,6 2EX-INTER-2OPT-INTRA-CSL

4,730.63 4,839.64 4,797.55 7,791.45 8,108.39 7,922.65

1,308,162,755.00 1,493,749,553.00 1,321,884,322.00

0.03 -0.37 -0.23

1.12 1.51 3.07

0.36 0.62 0.34 0.78

H5,5 2EX-INTER-2OPT-INTRA-3EX

7,755.43 8,045.47 7,796.68 20,509.33 21,419.95 20,779.72

1,413,223,585.00 1,522,717,925.00 1,440,865,895.00

1.82 1.84 2.57

-0.05 0.38 0.21 0.59

CE

Test instance

Table 17: Average solutions and average number of evaluations on the basis of 2EX-INTER-2OPT-CSL

small all with trailer all without trailer normal medium all with trailer all without trailer normal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal Average

Test instance

20,215.26 21,255.51 20,535.70 25,146.30 26,053.92 25,587.32

1,273,557,764.00 1,445,492,444.00 1,301,767,998.00

0.36 0.62 0.34 0.78

Gap (2, H4,1 )

25,138.40 26,149.90 25,647.00

34,632.11 37,932.78 35,714.54

685,375,858.00 1,382,526,882.00 697,351,469.00

Gap (2, H4,4 )

34,011.59 37,248.88 34,820.79

129,886.15 145,845.48 132,931.51

-0.07 0.40 -0.07

129,420.62 144,945.35 132,484.84

Table 18: Average solutions and average number of evaluations on the basis of 2EX-INTER-2OPT-INTRA

small all with trailer all without trailer normal medium all with trailer all without trailer normal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal Average

0.04 -0.40 -0.04 0.06

AC

24

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

25

and H4,4 since the average gap is only 0.35% respectively 0.19%. In addition, the average Gap (1, H3,4 ) is -0.17% and -0.07% for Gap (1, H3,1 ). The average gap can be further improved for the order H4,1 (Gap (4, H3,4 ) = -0.33%) and in case of H4,4 (Gap (4, H3,1 ) = -0.23%). In terms of the sequence H4,3 , it is shown that the results are superior to the sequences with 3 operators, namely the H3,4 /H3,1 . Taking these findings into account, both sequences, H4,1 and H4,4 , are tested and it is analyzed how an extra operator affects the solution quality. The results obtained based on the sequences H4,4 and H4,1 are summarized in Tables 17 and 18.

CR IP T

• Thereby, the Gap (l, H4,4 ) is the gap between the investigated average solution, H5,l with l = 1, . . . , 9, and the average solution of the sequence H4,4 (Gap (l, H4,4 ) = ((Avg. Sol.(H5,l ) - Avg. Sol.(H4,4 ))/Avg. Sol.(H4,4 ))*100). • Gap (l, H4,1 ) is defined as the gap between the average solution, H5,l , and the average solution of the sequence H4,1 (Gap (l, H4,1 ) = ((Avg. Sol.(H5,l ) Avg. Sol.(H4,1 ))/Avg. Sol.(H4,1 ))*100).

5.7

AN US

In Tables 17 and 18, it can be observed that the lowest average gap over all instances is reached by the sequence H5,1 (0.40%), followed by the sequences H5,6 (0.41%) and H5,2 (0.74%). Gap (H4,4 ) and Gap (H4,1 ) with l = 1, 2, 3, 4, 5, 6 has positive values which shows that an additional operator cannot improve the solutions. In these cases, it is not appropriate to analyze sequences with 6 neighborhood operators. The results for the orders with 5 neighborhood operators show that no sequence can outperform the results with only 4 neighborhood operators. With respect to these findings, the investigation of a sequence with 6 or more operators seems not beneficial.

Summary of the results

ED

M

To understand the behavior of the interaction effects of the proposed neighborhood operators, we have provided an experimental setting in order to solve the sequencingand the selection-problem. In fact, we analyze the tradeoff between the inclusion of a operator which utilizes limited CPU time and the solution quality. The incorporation of an additional operator does not necessarily lead to better results. We also explored that more sophisticated operators, such as the 3OPT/ 3EX, could not provide improved numerical values. It is important to note that these effects cannot be estimated before the conducted experiments.

AC

CE

PT

Table 19: Improved BKS over all experiments

small all with trailer all without trailer normal medium all with trailer all without trailer normal large1 all with trailer all without trailer normal large2 all with trailer all without trailer normal final all with trailer all without trailer normal final random all with trailer all without trailer normal Average

BKS

Best Sol. all experiments

Gap BKS

4,730.63 4,839.64 4,797.55

4,716.58 4,839.64 4,797.55

-0.30 0.00 0.00

7,755.43 8,045.47 7,796.68

7,734.61 7,982.76 7,795.98

-0.27 -0.78 -0.01

20,215.26 21,255.51 20,535.70

20,058.99 21,003.51 20,298.82

-0.77 -1.19 -1.15

25,138.40 26,149.90 25,647.00

24,767.63 25,719.19 25,069.86

-1.47 -1.65 -2.25

34,011.59 37,248.88 34,820.79

33,753.48 36,814.84 34,649.78

-0.76 -1.17 -0.49

129,420.62 144,945.35 132,484.84

129,257.44 144,725.57 132,295.68

-0.13 -0.15 -0.14 -0.70

Overall, we showed that the sequence with four operators, 2EX → INTER → 2OPT → CSL (H4,4 ) achieved, on average, the best results for all benchmark instances. These results highlight the absence of the INTRA, 3OPT and 3EX operators

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to obtain high quality solutions. Additionally, the promising sequence includes operators of each category of the classification scheme which is the inter-, intra- and problem-specific category. It is important to notice that the implemented operators are not independent of the problem structure. For our problem, we implemented operators which can modify the sequence of the customers as well as the clusters of the customers. Thus, it is not surprising that the sequence incorporates operators of both categories. However, it is not clear which operators of each category should be chosen. The performance of the algorithm clearly shows that the sequence matters, at least for the investigated problem. We are identifying a sequence that has an overall good performance for all instances. However, our proposed setting can be applied to e. g. every instance set in order to improve the solution quality. Promising sequences can be different for each test set due to the different characteristics, such as number of customers etc. For the small set, it is e. g. beneficial to include 3OPT and 3EX (see Table 15). In Table 19 the results are summarized with respect to the improved BKSs over all experiments. We see that 16 out of 18 BKSs could be improved and VNS is capable of equaling the results of the other 2 instances.

Additional benchmark instances

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Twenty new test instances have been published in the study of [1] and are available under: http://www.vrp-rep.org/datasets.html. A summary of the main characteristics is presented in Table 20. These new instances are obtained by adapting the maximal driving time and/ or the demand of the VeRoLog instances. In the second column of Table 20 the modifications are described for each class. Note that customers are removed which cannot be served by a round trip. Also, customers with a demand higher than 2Q are discarded. Table 20: Characteristics of the new instances [1] Modification

Class 1

Driving time is multiplied by two.

Class 2

Maximal time is divided by two.

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Class 4

Class 5

The demand is multiplied by two.

The maximal driving time and the demand of each customer is divided by two.

Size

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large1 large2 medium small large1 large2 medium small large1 large2 medium small large1 large2 medium small large1 large2 medium small

548 550 206 57 520 481 199 39 548 550 206 57 548 550 203 56 520 481 199 39

Experiments are conducted on the new benchmark instances. The solution method uses the previously determined sequence of the neighborhood operators, which is 2EX → INTER → 2OPT → CSL (H4,4 ). Also, the other parameters are the same, such as e. g. number of routes to remove. Computation time is 600 seconds as proposed in the challenge. However, the work of [1] utilizes in total 900 seconds. We continue with the given time limit of the challenge in order to be cautious with the numerical analysis. Results are presented in Table 21, where ‘our algorithm’ is compared with the results of [1]. Table 21 gives the name of the instances in the first column, followed

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Table 21: Comparison of the results. New BKSs found by our algorithm are underlined. Also, average values below the BKS are highlighted. Absi et al. [1] BKS

Avg. Sol. Best Sol. Avg. Sol. Avg. eval.

Gap BKS

20,008.50 21,084.30 7,411.30 4,292.07 37,954.90 50,052.80 14,450.50 3,437.63 17,606.80 22,806.50 6,809.87 3,975.87 30,183.10 33,601.20 11,015.70 6,950.81 37,338.10 50,054.60 14,329.80 3,222.84

20,246.73 21,328.52 7,532.78 4,302.80 38,031.15 50,076.43 14,457.48 3,444.68 17,776.52 22,874.95 6,820.14 3,975.87 30,409.00 33,835.42 11,072.40 6,981.30 37,410.13 50,066.77 14,335.62 3,222.84

-5.98 -0.90 -1.61 0.39 0.84 0.08 -0.02 1.23 -0.19 0.13 0.14 0.00 -4.28 -5.96 -1.11 -0.23 1.08 0.17 0.06 1.53 -0.73

19,031.76 21,153.84 7,396.62 4,357.08 38,409.30 50,314.54 14,490.59 3,479.94 17,715.46 23,055.37 6,879.43 3,975.87 29,071.78 31,770.11 10,950.33 6,934.71 37,878.22 50,313.94 14,351.06 3,272.11

1,121,451,427 1,096,693,983 1,176,547,287 1,475,377,392 1,956,840,337 2,058,438,660 1,871,587,393 2,006,247,955 1,398,205,476 1,538,331,225 1,446,241,188 1,892,335,396 1,251,170,433 1,269,681,610 1,348,442,304 1,517,391,163 1,980,355,963 2,056,750,914 1,874,908,091 2,062,913,527

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18,812.41 20,894.22 7,292.13 4,309.01 38,273.61 50,093.70 14,447.51 3,479.94 17,573.49 22,835.79 6,819.36 3,975.87 28,890.85 31,596.90 10,893.66 6,934.71 37,740.84 50,139.53 14,338.43 3,272.11

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C1 large1 C1 large2 C1 medium C1 small C2 large1 C2 large2 C2 medium C2 small C3 large1 C3 large2 C3 medium C3 small C4 large1 C4 large2 C4 medium C4 small C5 large1 C5 large2 C5 medium C5 small Average

Our algorithm

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by the BKSs and average solutions of [1]. The fourth respectively the fifth column shows the best solution/ average solution of our algorithm (30 runs). In the last two columns, the number of evaluations and the Gap with respect to the BKS is specified. Without having any additional information on the test instances, our algorithm is capable of solving the instances. Overall, the average Gap BKS is −0.73. In total, nine new BKSs can be found (C1 large1, C1 large2, C1 medium, C2 medium, C3 large1, C4 large1, C4 large2, C4 medium and C4 small) and one BKS is equaled (C3 small). Note that in every run the same objective value is computed for C3 small. In addition, the average solution values are highlighted which are below the BKS. This behaviour is observed for the following instances: C1 large1, C4 large1, C4 large2, C4 medium and C4 small. Similar to the idea of [15], we analyze the solution quality relative to the running time. Exemplarily, the progress of the algorithm is analyzed in the Tables 22 to 25 and Figures 9 to 16. Figure 9 illustrates the average progress (30 runs) of our algorithm during the running time. Our algorithm reaches the BKS on average after only 29 seconds for instance C1 large1. For instance C1 large2, the proposed solution method improves the BKS. However, the average progress shows a small average Gap to the BKS of 0.33 (see Figure 10). A detailed overview of the average solution values, the standard deviation (std. dev.) and the average Gap to the BKSs is presented in Table 22. In comparison to the average results for the C1 medium and C1 small instances (Table 23), the standard deviation is larger. This is simply due to the fact that the C1 large1/ C1 large2 instance is bigger than the medium/ small instance. Similar to the behaviour for the C1 large1 instance, the BKS can be found on average after 387 seconds for instance C1 medium (see Figure 11). For the C1 small instance, our algorithm has on average a small Gap to the BKS of 1.51%. We observe that our algorithm is relatively fast for the C4 data sets. The BKS is surpassed on average after only 26 seconds for instance C4 large1 (see Figure 13 and Table 24) and even faster for instance C4 large2, where the BKS is surpassed after 19 seconds on average (Figure 14). From Figure 15 as well as from Figure 16, we can

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Table 22: Average results for C1 large1 (left) and C1 large2 (right) Avg. Sol.

Std. dev.

Avg. Gap BKS

seconds

Avg. Sol.

Std. dev.

Avg. Gap BKS

10 30 60 120 180 240 300 360 420 480 540 600

21,165.56 19,991.61 19,556.21 19,319.66 19,233.68 19,184.54 19,139.24 19,112.17 19,092.11 19,072.99 19,054.13 19,031.76

76.38 217.47 177.90 119.50 104.62 100.70 102.04 104.23 99.25 102.82 100.85 100.50

5.78 -0.08 -2.26 -3.44 -3.87 -4.12 -4.34 -4.48 -4.58 -4.68 -4.77 -4.88

10 30 60 120 180 240 300 360 420 480 540 600

23,462.32 22,360.83 21,861.04 21,583.35 21,463.54 21,379.43 21,319.28 21,268.29 21,234.54 21,203.09 21,177.86 21,153.84

154.12 188.79 155.82 192.79 195.03 194.45 180.93 167.58 158.55 164.96 159.23 152.62

11.28 6.05 3.68 2.37 1.80 1.40 1.11 0.87 0.71 0.56 0.44 0.33

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Table 23: Average results for C1 medium (left) and C1 small (right) Std. dev.

Avg. Gap BKS

seconds

Avg. Sol.

Std. dev.

Avg. Gap BKS

7,694.84 7,544.57 7,495.50 7,456.69 7,438.15 7,423.95 7,418.12 7,412.15 7,408.44 7,404.75 7,399.11 7,396.62

59.70 83.18 76.50 69.36 68.70 65.85 64.54 63.58 63.59 63.88 62.50 63.88

3.83 1.80 1.14 0.61 0.36 0.17 0.09 0.01 -0.04 -0.09 -0.16 -0.20

10 30 60 120 180 240 300 360 420 480 540 600

4,372.75 4,362.38 4,362.35 4,362.31 4,362.31 4,362.28 4,362.21 4,361.53 4,357.29 4,357.08 4,357.08 4,357.08

7.78 0.06 0.00 0.08 0.08 0.11 0.11 3.65 14.77 15.41 15.41 15.41

1.88 1.64 1.64 1.64 1.64 1.64 1.63 1.62 1.52 1.51 1.51 1.51

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21,000 20,500 20,000 19,500 19,000

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Figure 9: Average progress (30 runs) of the solution quality during the running time for instance C1 large1. Light grey horizontal line: BKS; dark grey line: average quality of the implementation with sequence 2EX → INTER → 2OPT → CSL. 24,500 24,000 23,500

22,500

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Figure 10: Average progress of the solution quality during the running time for instance C1 large2. Light grey horizontal line: BKS; dark grey line: average quality of the implementation with sequence 2EX → INTER → 2OPT → CSL.

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analyze that the algorithm is able to reach the BKS on average after only 113 seconds (instance C4 medium) and on average after only 9 seconds for instance C4 small. As presented in Table 25, the results are robust since the standard deviation for instance C4 small is 0. Based on the results, we derive that our algorithm performs especially well on the instances of class C1 and on the class C4. The solution approach can handle the two characteristics, such as 1) the multiplication of the driving time by two and 2) the multiplication of the demand by two, better than the others. In Table 26, we compare the different instances with respect to the minimal and the maximal number of routes, the average vehicle-utilization and the average time-utilization. For the C1 as well as the C4 instances the average vehicle-utilization is relatively high. In the case of the C1/ C4 instances, it is difficult to include additional customers in the routes since the utilization has high values with around 98% respectively 97%. For the C2/

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8,100 8,000 7,900

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Figure 11: Average progress of the solution quality during the running time for instance C1 medium. Light grey horizontal line: BKS; dark grey line: average quality of the implementation with sequence 2EX → INTER → 2OPT → CSL.

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4,420 4,400

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4,360 4,340

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4,320 4,300 4,280

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Figure 12: Average progress of the solution quality during the running time for instance C1 small. Light grey horizontal line: BKS; dark grey line: average quality of the implementation with sequence 2EX → INTER → 2OPT → CSL.

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32,000 31,500

30,500 30,000 29,500 29,000 28,500 1

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Figure 13: Average progress of the solution quality during the running time for instance C4 large1. Light grey horizontal line: BKS; dark grey line: average quality of the implementation with sequence 2EX → INTER → 2OPT → CSL.

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35,500 35,000

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34,500

33,500

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34,000

33,000

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32,500 32,000 31,500

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Std. dev.

Avg. Gap BKS

seconds

Avg. Sol.

Std. dev.

Avg. Gap BKS

10 30 60 120 180 240 300 360 420 480 540 600

31,379.77 30,229.70 29,810.86 29,468.35 29,332.89 29,266.05 29,198.84 29,164.40 29,137.52 29,110.72 29,088.98 29,071.78

49.15 165.77 133.22 123.31 113.57 124.66 105.96 108.50 104.51 101.87 108.54 109.60

3.19 -0.59 -1.97 -3.09 -3.54 -3.76 -3.98 -4.09 -4.18 -4.27 -4.34 -4.40

10 30 60 120 180 240 300 360 420 480 540 600

34,563.99 33,330.79 32,708.43 32,308.81 32,135.84 32,028.04 31,945.34 31,888.35 31,847.34 31,817.06 31,790.96 31,770.11

205.50 170.74 148.00 157.18 156.92 140.42 143.91 136.18 140.72 137.92 119.69 119.27

2.15 -1.49 -3.33 -4.51 -5.02 -5.34 -5.59 -5.75 -5.88 -5.97 -6.04 -6.10

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11,800 11,700

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11,500 11,400 11,300 11,200

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Figure 15: Average progress of the solution quality during the running time for instance C4 medium. Light grey horizontal line: BKS; dark grey line: average quality of the implementation with sequence 2EX → INTER → 2OPT → CSL. Table 25: Average results for C4 medium (left) and C4 small (right)

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seconds 10 30 60 120 180 240 300 360 420 480 540 600

Avg. Sol.

Std. dev.

Avg. Gap BKS

seconds

Avg. Sol.

Std. dev.

Avg. Gap BKS

11,230.28 11,115.83 11,051.42 11,014.13 10,998.88 10,986.66 10,976.79 10,972.36 10,963.33 10,960.67 10,955.87 10,950.33

6.19 30.75 38.27 32.15 34.13 35.70 30.43 30.97 32.26 32.66 34.88 34.79

1.95 0.91 0.32 -0.01 -0.15 -0.26 -0.35 -0.39 -0.48 -0.50 -0.54 -0.59

10 30 60 120 180 240 300 360 420 480 540 600

6,935.39 6,935.39 6,935.39 6,934.96 6,934.96 6,934.96 6,934.71 6,934.71 6,934.71 6,934.71 6,934.71 6,934.71

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

-0.22 -0.22 -0.22 -0.23 -0.23 -0.23 -0.23 -0.23 -0.23 -0.23 -0.23 -0.23

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6,985 6,980 6,975 6,970

Costs

6,965 6,960 6,955 6,945 6,940 6,935 6,930 1

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Figure 16: Average progress of the solution quality during the running time for instance C4 small. Light grey horizontal line: BKS; dark grey line: average quality of the implementation with sequence 2EX → INTER → 2OPT → CSL.

Table 26: Comparison of the average results with respect to the avg. number of vehicles, avg. vehicle utilization % and avg. time utilization % Avg. timeutilization %

36 33 13 9 125 161 48 14 41 50 16 10 70 66 26 17 125 161 48 14

38 36 15 10 127 162 48 14 42 51 16 10 74 69 27 17 126 162 48 14

98.58 98.01 98.95 94.16 46.39 30.64 46.60 66.43 74.31 57.85 72.64 74.97 97.73 95.76 97.11 94.76 25.00 15.32 25.16 41.91

53.14 64.59 53.39 47.18 96.25 96.76 96.55 90.53 96.59 97.44 96.29 95.50 65.38 81.84 66.13 74.73 96.71 96.70 96.36 90.13

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large1 large2 medium small large1 large2 medium small large1 large2 medium small large1 large2 medium small large1 large2 medium small

Minimal no of routes

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C1 C1 C1 C1 C2 C2 C2 C2 C3 C3 C3 C3 C4 C4 C4 C4 C5 C5 C5 C5

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C3/ C5 instances the average time-utilization seems more important. Different to the previous cases, further customers cannot be included in the route since the driver has not enough time. Note that for instance C2 medium, C2 small, C3 medium, C3 small, C4 small, C5 medium and C5 small, the algorithm always computes solutions with the same number of vehicles.

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Conclusions

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References

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In this paper the performance of the neighborhood operators has been investigated for the SB-VRP. The analysis of the solely applied operators showed that the two inter-tour-operators 2-EX and INTER performed remarkable better than the CSL, INTRA, 2OPT, 3OPT and the 3EX. We also observed that this rather simplistic idea with only one neighborhood operator achieves better results than the studies of [33, 34]. This is in line with the investigation of [37] which shows that “less is more” by using only one operator, resulting in highly competitive results, where new best solutions are found. Despite this good results with only one neighborhood operator, the impact of including an additional operator is further analyzed by fixing the first position with the 2EX respectively the INTER operator and incorporating the other operators on the second position. The procedure is continued until a point, where an extra neighborhood operator in the sequence is not beneficial. This choice problem occurs since the optimization time is scarce and using an additional operator also means that this computational time can not be utilized by the other operators. Another contribution is made by the investigation on the parallel execution of the Variable Neighborhood Searches. An efficient parallelization is becoming more important since practical problems are getting larger and in addition, often more computational power is available [28]. The experiments carried out on the synchronisation times of the master-worker model showed that lower synchronisation times, resulting in more information exchanges, yield to overall better results at least for the investigated instances. Future research can investigate the effects on the solution quality when e. g. different promising sequences are implemented in parallel.

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[1] Absi, N., Cattaruzza, D., Feillet, D., Housseman, S.: A relax-and-repair heuristic for the Swap-Body Vehicle Routing Problem. Annals of Operations Research pp. 1–22 (2015)

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[2] Azi, N., Gendreau, M., Potvin, J.Y.: An adaptive large neighborhood search for a vehicle routing problem with multiple routes. Computers & Operations Research 41, 167–173 (2014)

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[3] B¨ogl, M., Z¨apfel, G., Affenzeller, M.: Computational study of neighborhood operator performance on the traveling salesman problem with time windows in neighborhood search based frameworks (RTS, VNS). In: 3rd IEEE International Symposium on Logistics and Industrial Informatics (LINDI), pp. 39–44 (2011) [4] Br¨aysy, O., Gendreau, M.: Vehicle Routing Problem with Time Windows, Part I: Route Construction and Local Search Algorithms. Transportation Science 39(1), 104–118 (2005) [5] Cafieri, S., Hansen, P., Mladenovi´c, N.: Edge-ratio network clustering by Variable Neighborhood Search. The European Physical Journal B 87(5), 1–7 (2014)

[6] Cheikh, M., Ratli, M., Mkaouar, O., Jarboui, B.: A variable neighborhood search algorithm for the vehicle routing problem with multiple trips. Electronic Notes in Discrete Mathematics 47, 277–284 (2015)

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[7] Cuervo, D.P., Goos, P., S¨orensen, K., Arr´aiz, E.: An iterated local search algorithm for the vehicle routing problem with backhauls. European Journal of Operational Research 237(2), 454–464 (2014) [8] Dang, N.T.T., De Causmaecker, P.: Characterization of Neighborhood Behaviours in a Multi-neighborhood Local Search Algorithm. In: P. Festa, M. Sellmann, J. Vanschoren (eds.) Learning and Intelligent Optimization: 10th International Conference, LION 10, Ischia, Italy, May 29 – June 1, 2016, Revised Selected Papers, pp. 234–239. Springer International Publishing (2016)

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[9] de Armas, J., Lalla-Ruiz, E., Exp´osito-Izquierdo, C., Landa-Silva, D., Meli´anBatista, B.: A hybrid GRASP-VNS for ship routing and scheduling problem with discretized time windows . Engineering Applications of Artificial Intelligence 45, 350–360 (2015) [10] Derigs, U., Pullmann, M., Vogel, U.: Truck and trailer routing – Problems, heuristics and computational experience. Computers & Operations Research 40(2), 536–546 (2013) [11] Drexl, M.: Branch-and-price and heuristic column generation for the generalized truck-and-trailer routing problem. Journal of Quantitative Methods for Economics and Business Administration 12(1), 5–38 (2011)

AN US

[12] Drexl, M.: Synchronization in vehicle routing – A survey of VRPs with multiple synchronization constraints. Transportation Science 46(3), 297–316 (2012) [13] Drexl, M.: Applications of the vehicle routing problem with trailers and transshipments. European Journal of Operational Research 227(2), 275–283 (2013)

[14] Geiger, M.J.: On heuristic search for the single machine total weighted tardiness problem – some theoretical insights and their empirical verification. European Journal of Operational Research 207(3), 1235–1243 (2010)

M

[15] Geiger, M.J.: On an effective approach for the coach trip with shuttle service problem of the VeRoLog Solver Challenge 2015. Networks (2017)

ED

[16] Glover, F.: Ejection chains, reference structures and alternating path methods for traveling salesman problems. Discrete Applied Mathematics 65(1-3), 223–253 (1996)

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[17] G¨ und¨ uz, H.I.: The single-stage location-routing problem with time windows. In: J.W. B¨ose, H. Hu, C. Jahn, X. Shi, R. Stahlbock, S. Voß (eds.) Proceedings of the 2nd International Conference on Computational Logistics (ICCL 2011), LNCS 6971, pp. 44–58. Springer, Berlin (2011)

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[18] Gust-Kazakos, P.: VeRoLog Solver Challenge 2014 (VSC2014): Ausgezeichnete L¨osungen (2014). URL http://compass.ptvgroup.com/tag/ verolog-solver-challenge-2014-vsc2014/

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[19] Hansen, P., Mladenovi´c, N.: First vs. best improvement: An empirical study. Discrete Applied Mathematics 154(5), 802–817 (2006)

[20] Hansen, P., Mladenovi´c, N.: Variable Neighborhood Search. In: E.K. Burke, G. Kendall (eds.) Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques, pp. 313–337. Springer US (2014)

[21] Hansen, P., Mladenovi´c, N., Brimberg, J., P´erez, J.A.M.: Variable Neighborhood Search. In: M. Gendreau, J.Y. Potvin (eds.) Handbook of Metaheuristics, International Series in Operations Research & Management Science, vol. 146, pp. 61–86. Springer US (2010) [22] Hansen, P., Mladenovi´c, N., Moreno P´erez, J.A.: Variable neighbourhood search: methods and applications. Annals of Operations Research 175(1), 367–407 (2010)

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[23] Heid, W., Hasle, G., Vigo, D.: Verolog solver challenge 2014 – VSC2014 problem description (2014). URL {http://verolog.deis.unibo.it/news-events/ general-news/verolog-solver-challenge-2014} [24] Heid, W., Hasle, G., Vigo, D.: Verolog solver challenge 2014 final results (2014). URL http://verolog.deis.unibo.it/news-events/general-news/ verolog-solver-challenge-2014-final-results [25] Hemmelmayr, V.C., Doerner, K.F., Hartl, R.F.: A variable neighborhood search heuristic for periodic routing problems. European Journal of Operational Research 195(3), 791–802 (2009)

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[26] Huber, S., Geiger, M.J.: Swap body vehicle routing problem: A heuristic solution approach. In: R.G. Gonz´alez-Ram´ırez, F. Schulte, S. Voß, J.A. Ceroni D´ıaz (eds.) Computational Logistics, Lecture Notes in Computer Science, vol. 8760, pp. 16– 30. Springer International Publishing (2014)

[27] Huber, S., Geiger, M.J.: Dealing with scarce optimization time in complex logistics optimization: A study on the biobjective swap-body inventory routing problem. In: A. Gaspar-Cunha, C. Henggeler Antunes, C.C. Coello (eds.) Evolutionary Multi-Criterion Optimization, Lecture Notes in Computer Science, vol. 9019, pp. 279–294. Springer International Publishing (2015)

AN US

ˇ [28] Jeannot, E., Zilinskas, J. (eds.): High-Performance Computing on Complex Environments. Wiley Series on Parallel and Distributed Computing. Wiley, USA (2014) [29] Juan, A.A., Faulin, J., Jorba, J., Caceres, J., Marqu`es, J.M.: Using parallel & distributed computing for real-time solving of vehicle routing problems with stochastic demands. Annals of Operations Research 207(1), 43–65 (2013)

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[30] Lin, S.: Computer solutions of the traveling salesman problem. Bell System Technical Journal 44(10), 2245–2269 (1965)

ED

[31] Lin, S.W., Yu, V.F., Chou, S.Y.: Solving the truck and trailer routing problem based on a simulated annealing heuristic. Computers & Operations Research 36(5), 1683–1692 (2009)

PT

[32] Louren¸co, H.R., Martin, O.C., St¨ utzle, T.: Iterated Local Search: Framework and Applications. In: M. Gendreau, J.Y. Potvin (eds.) Handbook of Metaheuristics, International Series in Operations Research & Management Science, vol. 146, pp. 363–397. Springer (2010)

CE

[33] Lum, O., Chen, P., Wang, X., Golden, B., Wasil, E.: A heuristic approach for the swap-body vehicle routing problem. In: 14th INFORMS Computing Society Conference, pp. 172–187 (2015)

AC

[34] Miranda-Bront, J.J., Curcio, B., M´endez-D´ıaz, I., Montero, A., Pousa, F., Zabala, P.: A cluster-first route-second approach for the swap body vehicle routing problem. Annals of Operations Research pp. 1–22 (2016) [35] Mirmohammadsadeghi, S., Ahmed, S.: Memetic Heuristic Approach for Solving Truck and Trailer Routing Problems with Stochastic Demands and Time Windows. Networks and Spatial Economics 15(4), 1093–1115 (2015) [36] Mladenovi´c, N., Hansen, P.: Variable neighborhood search. Computers & Operations Research 24(11), 1097–1100 (1997) [37] Mladenovi´c, N., Todosijevi´c, R., Uroˇsevi´c, D.: Less is more: Basic variable neighborhood search for minimum differential dispersion problem. Information Sciences 326, 160–171 (2016)

[38] Mladenovi´c, N., Uroˇsevi´c, D., Hanafi, S., Ili´c, A.: A general variable neighborhood search for the one-commodity pickup-and-delivery travelling salesman problem. European Journal of Operational Research 220(1), 270–285 (2012)

ACCEPTED MANUSCRIPT

37

[39] Penna, P.H.V., Subramanian, A., Ochi, L.S.: An iterated local search heuristic for the heterogeneous fleet vehicle routing problem. Journal of Heuristics 19(2), 201–232 (2013) [40] P´erez, J.A.M., Hansen, P., Mladenovi´c, N.: Parallel variable neighborhood search. In: E. Alba (ed.) Parallel Metaheuristics: A New Class of Algorithms, Wiley Series on Parallel and Distributes Computing, pp. 247–266. Wiley (2005) [41] Puchinger, J., Raidl, G.R.: Bringing order into the neighborhoods: relaxation guided variable neighborhood search. Journal of Heuristics 14(5), 457–472 (2008) [42] Scheuerer, S.: A tabu search method for the truck and trailer routing problem. Computers & Operations Research 33(4), 894–909 (2006)

CR IP T

[43] Scholz-Reiter, B., Hildebrandt, T., Tan, Y.: Effective and efficient scheduling of dynamic job shops – Combining the shifting bottleneck procedure with variable neighbourhood search. CIRP Annals - Manufacturing Technology 62(1), 423–426 (2013) [44] Schulte, F., Voß, S., Wenzel, P.: Heuristic routing software for planning of combined road transport with swap bodies: A practical case. In: Proceedings of MKWI 2014 – Multikonferenz Wirtschaftsinformatik, 15-28 February 2014, pp. 1513–1524. Paderborn (2014)

AN US

[45] Silva, M.M., Subramanian, A., Ochi, L.S.: An iterated local search heuristic for the split delivery vehicle routing problem. Computers & Operations Research 53, 234–249 (2015)

[46] Soylu, B.: A general variable neighborhood search heuristic for multiple traveling salesmen problem. Computers & Industrial Engineering 90, 390–401 (2015)

M

[47] Subramanian, A., Drummond, L.M.A., Bentes, C., Ochi, L.S., Farias, R.: A parallel heuristic for the vehicle routing problem with simultaneous pickup and delivery. Computers & Operations Research 37(11), 1899–1911 (2010)

ED

[48] Subramanian, A., Ochi, L.S., Cabral, L.d.A.F.: An efficient ILS heuristic for the Vehicle Routing Problem with Simultaneous Pickup and Delivery. Tech. rep., Universidade Federal Fluminense (2008). URL http://www2.ic.uff.br/ PosGraduacao/RelTecnicos/401.pdf

PT

[49] Talarico, L., S¨orensen, K., Springael, J.: The k-dissimilar vehicle routing problem. European Journal of Operational Research 244(1), 129–140 (2015)

CE

[50] Tan, K.C., Chew, Y.H., Lee, L.H.: A hybrid multiobjective evolutionary algorithm for solving truck and trailer vehicle routing problems. European Journal of Operational Research 172(3), 855–885 (2006) [51] Todosijevi´c, R., Hanafi, S., Uroˇsevi´c, D., Jarboui, B., Gendron, B.: A general variable neighborhood search for the swap-body vehicle routing problem. Computers & Operations Research (2016)

AC

[52] Villegas, J.G., Prins, C., Prodhon, C., Medaglia, A.L., Velasco, N.: A matheuristic for the truck and trailer routing problem. European Journal of Operational Research 230(2), 231–244 (2013) [53] Wauters, T., Toffolo, T.A.M., Christiaens, J., Van Malderen, S.: The winning approach for the Verolog Solver Challenge 2014: the Swap-Body Vehicle Routing Problem. Proceedings of the 29th Belgian Conference on Operations Research (ORBEL29) (2015) [54] Yahyaoui, H., Krichen, S.: A DSS Based on Hybrid Meta-Heuristic ILS-VND for Solving the 1-PDTSP. In: L. Rutkowski, M. Korytkowski, R. Scherer, R. Tadeusiewicz, L.A. Zadeh, J.M. Zurada (eds.) Artificial Intelligence and Soft Computing, Lecture Notes in Computer Science, vol. 9120, pp. 789–798. Springer International Publishing (2015)