An evolution strategy approach to the team orienteering problem with time windows

Computers & Industrial Engineering 139 (2020) 106109

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An evolution strategy approach to the team orienteering problem with time windows

T

Korhan Karabuluta, , M. Fatih Tasgetirenb ⁎

a b

Software Engineering Department, Yasar University, Selcuk Yasar Campus, 35100 Izmir, Turkey Mechanical and Industrial Engineering Department, Qatar University, Doha, Qatar

ARTICLE INFO

ABSTRACT

Keywords: Evolution strategy Team orienteering problem with time windows Ruin and recreate heuristic

The team orienteering problem with time windows (TOPTW) is a highly constrained NP-hard problem having many practical applications in vehicle routing and production scheduling. The TOPTW is an extended variant of the Orienteering Problem (OP), where each node has a predefined time window during which the service has to be started. The aim is to maximize the total collected score by visiting a set of nodes with a limited number of tours since the given distance budget is limited. We propose an evolution strategy (ES) together with an effective constructive heuristic for solving the TOPTW. The main feature of the ES is to generate an offspring solution through ruin and recreate (RR) heuristic, where a number of nodes are removed from the incumbent solution and then, they are reinserted into tours until a complete solution is obtained. The ES is hybridized with an efficient random local search to enhance solution quality. For survivor selection, we use a goodness of scores approach to determine and diversify the population for the next generation. Parameters of the ES are determined through the design of experiment approach to tune them. The computational results show that the constructive heuristic is slightly better than existing heuristics in the literature. Furthermore, the detailed computation results on the benchmark suite from the literature confirm the effectiveness of the evolution strategy. Ultimately, the evolution strategy obtains new best-known solutions for 7 benchmark problem instances.

1. Introduction and literature review As a sports discipline, orienteering can be defined as follows. Given a set of control points with associated profits together with the start and end points, the orienteering problem (OP) aims to obtain a tour between the start and endpoints so as to maximize the total profit collected; subject to a given distance budget. Since distance is limited, a tour cannot include all points. The first public orienteering competition was held in 1897 in Norway; however, the practice in the army is earlier (IOF, n.d.). Orienteering was first introduced as an optimization problem by Tsiligirides (1984). Then, Golden, Levy, and Vohra (1987) presented a heuristic algorithm to solve the OP in three steps: route construction, route improvement, and center of gravity improvement. Golden, Wang, and Liu (1988) designed a new and improved heuristic for the OP. The randomization concept of Tsiligrides is embedded in the center of gravity procedure together with learning capabilities. Keller (1989) proposed an algorithm for a multi-objective vending problem to solve the OP. His algorithm consists of two stages: path construction and improvement. Ramesh and Brown (1991) developed an efficient four-

⁎

phase heuristic for the OP. Their heuristic consists of vertex insertion, cost improvement, vertex deletion, and maximal insertions. Chao (1993) and Chao, Golden, and Wasil (1996) developed a fast and effective heuristic for the OP. Tasgetiren and Smith (2000) and Tasgetiren (2002) developed genetic algorithms for the OP. The OP can be described as a combination of the 0–1 knapsack and the TSP (Vansteenwegen, Souffriau, & Oudheusden, 2011). Certain problems in logistics, tourism and other areas can be modeled as an orienteering problem (Vansteenwegen et al., 2011). There exist several variants of the OP in the literature. One version is called the OP with time windows (OPTW), where a time window refers to the practical situation that control points should be visited within a predefined time interval to start the service. This additional constraint makes finding feasible solutions harder. The literature on the OPTW is very scarce. Kantor and Rosenwein present a heuristic based on exhaustive search (1992), Mansini, Pelizzari, and Calvo (2006) have designed a granular variable neighborhood search, and Righini and Salani (2009) proposed a dynamic programming approach. Santini (2019) proposed a heuristic that uses clustering and an algorithm based on adaptive large neighborhood search.

Corresponding author. E-mail addresses: [email protected] (K. Karabulut), [email protected] (M.F. Tasgetiren).

https://doi.org/10.1016/j.cie.2019.106109 Received 9 April 2019; Received in revised form 24 June 2019; Accepted 1 October 2019 Available online 26 October 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved.

Computers & Industrial Engineering 139 (2020) 106109

K. Karabulut and M.F. Tasgetiren

Another variant is the team orienteering problem (TOP), which is proposed by Chao et al. (1996a, 1996b). In the TOP, a fixed number m of paths is considered with a given time budget for each path. Chao et al. (1996a, 1996b) employed a two-phase heuristic algorithm, originally designed for the OP (Chao, 1993) to solve the TOP. A tabu search is presented by Tang and Miller-Hooks (2005) to solve the TOP. An ant colony optimization (ACO) algorithm is proposed by Ke, Archetti, and Feng (2008). A memetic algorithm is presented by Bouly, Dang, and Moukrim (2010). Souffriau, Vansteenwegen, Berghe, and Van Oudheusden (2010) develop a path-relinking algorithm combined with a greedy randomized adaptive search procedure. Vansteenwegen et al. also present a guided local search and a skewed variable neighborhood search for the TOP (2009a, 2009b, 2009c). The TOP with time windows (TOPTW) is an extended variant of the TOP, where each node has a predefined time window during which the service has to be started. The aim is to maximize the total collected score by visiting a set of nodes with a limited number of tours since the given distance budget is limited. Since the OP is NP-hard (Golden et al., 1987), it is unlikely to solve the TOPTW to optimality within polynomial time. All solution approaches for the TOPTW in the literature are metaheuristics, except for a few exact methods. An ant colony system (ACS) designed to solve the hierarchic generalization of the TOPTW is presented by Montemanni and Gambardella (2009). It is further enhanced by Montemanni, Weyland, and Gambardella (2011) by developing enhanced ACS (EACS). The EACS algorithm proposes two more components to handle the drawbacks of ACS. Both components employ the best solution found so far in the construction phase ASC and local search is applied to those solutions on which local search has not been recently applied. An iterated local search (ILS) algorithm is developed by Vansteenwegen et al. (2009a, 2009b, 2009c), which can provide fast solutions to the TOPTW. The proposed ILS is very simple and effective in a way that the current solution is perturbed and insertion based local search is applied to the perturbed solution. Tricoire, Romauch, Doerner, and Hartl (2010) developed a variable neighborhood search (VNS) algorithm with several neighborhood structures to solve the multi-period orienteering problem with multiple time windows, which is another generalization of the TOPTW, where visits are extended to multiple time-periods instead of a single period. A hybrid method consisting of a greedy randomized adaptive search procedure (GRASP) with evolutionary local search (ELS) is presented by Labadie, Melechovský, and Calvo (2011). Several constructive heuristics based on GRASP are developed for establishing the initial population. Then, those solutions in the initial population are further enhanced by ELS. Lin and Yu (2012) developed simulated annealing (SA)-based two heuristics called “slow SA (SSA)” and “fast SA (FSA)”. FSA is aimed at getting quick responses, whereas SSA is developed for getting highquality solutions at the expense of more CPU time. Labadie, Mansini, Melechovský, and Calvo (2012) developed a linear programming-based granular variable neighborhood search algorithm that can explore fewer neighborhoods instead of complete ones, without losing the effectiveness of the algorithm. Souffriau, Vansteenwegen, Vanden Berghe, and Van Oudheusden (2011) developed a hybrid GRASP and ILS algorithm named GRILS to newly proposed multi constraint team orienteering problem with multiple time windows as well as the TOPTW and selective vehicle routing problem with time windows. Hu and Lim (2014) developed an iterative framework including three components, a local search procedure, a simulated annealing procedure and a route recombination method (I3CH). Artificial bee colony (ABC) approach is first presented by Karabulut and Tasgetiren (2013). Later on, Cura (2014) also developed an ABC algorithm, where the hybridization of SA and a new scout bee search behavior based on a local search procedure are introduced to improve the solution quality of benchmark instances. Ke, Guo, and Zhang (2014) present an approach that uses both a metaheuristic and a branch-and-price algorithm whereas Gedik, Kirac, Milburn, and Rainwater (2017) presented a constraint programming (CP) approach to tackle the TOPTW. Very

recently, Gunawan, Lau, Vansteenwegen, and Lu (2017) improved the GRILS approach and present two new algorithms, i.e. iterated local search (ILS), and a hybrid Simulated Annealing ILS (SAILS) that are executed using different computation times in order to compare to the state-of-the-art algorithms. Gavalas, Konstantopoulos, Mastakas, and Pantziou (2019) proposed clustering-based heuristics to extend the ILS algorithm that could improve solution quality with respect to ILS. Yu, Jewpanya, Lin, and Redi (2019) used a hybrid ABC to solve TOPTWTDS which extends TOPTW by adding time-dependent scores where the profit of a location is dependent on the time of visit. They also report results of their algorithm for the TOPTW instances. A multi-objective TOPTW is proposed by Hu, Fathi, and Pardalos (2018) where different kinds of profits can be assigned to each checkpoint. Two exact methods for the OPTW (a special case of the TOPTW with only one tour) using bi-directional dynamic programming are presented by Righini and Salani (2006, 2009) that can find optimal results for the OPTW instances. To date, the only exact algorithm for the TOPTW, which is based on a branch-and-price approach, is given by Tae and Kim (2015) that can solve some of the TOPTW benchmark instances to optimality. A survey is first presented in Vansteenwegen et al. (2011). More details about the performances of the state-of-the-art algorithms for the TOPTW can be found in a recent survey in Gunawan, Lau, and Vansteenwegen (2016). In addition, practical applications of the TOPTW can be found in Souffriau, Vansteenwegen, Vertommen, Berghe, and Oudheusden (2008), Vansteenwegen and Oudheusden (2007), and Wang, Golden, and Wasil (2008). For the description of the TOPTW, we follow Labadie et al. (2012) as follows. Suppose that a set of nodes to be visited N = {1, 2, ..,n} is given with a depot denoted by 0. Tours always start and end at the depot. Assume that a copy of the depot is also indexed by n + 1. A directed graph can be defined as G = (N {0, n + 1}, E ) , where N {0, n + 1} is the set of nodes and E is a set of edges connecting the nodes such that E = {(i, j): i j N {0, n + 1}} . Each node i N has a profit Pi and each node can be visited at most once, whereas there is no profit for visiting the depot, i.e., P0 and Pn+ 1 = 0 . Each node i N has a time window [ei , li], in which ei is the earliest visiting time and li is the latest visiting time allowed to start the service at nodei . On the other hand, e0 and l 0 are the earliest visiting time each tour to depots 0 and n + 1. There exist m tours, each limited by the maximum tour length Tmax . We assume that e0 = en + 1 = 0 and l 0 = ln + 1 = Tmax . There is a cost cij in time units for travelling between nodes i and j and each node i has a service time denoted by Si . The problem is to find m tours with a maximum total profit. Each tour starts at time e0 , visits a subset of nodes exactly once within their time windows and ends at the depot n + 1 before the time l 0 = ln + 1 = Tmax . Let Li be the departure time from node i after its service is finished. With L0 = 0 , Li can be calculated as follows:

Li = max(Li

1

+ ci

1, i,

ei )+Si,

(1)

Let Ai be the arrival time at node i that can be calculated as follows:

Ai = Li

1

+ ci

(2)

1, i.

Then, the start of service time for node i can be calculated as follows:

SSTi = max(Li

1

+ ci

1, i,

ei)

(3)

A feasible solution to the TOPTW requires that the service for node i should start before its latest visiting time (SSTi li ) and arrival time to depot should be smaller than or equal to the maximum tour length ( An + 1 Tmax ) for all m tours. By the definitions above, a feasibility check of a solution can be easily carried out when making any move in any tour. Note that we only consider feasible moves in this study. In this paper, an effective constructive heuristic, which is denoted as KT and an evolution strategy (ES) are presented for the TOPTW. The KT heuristic is compared to best-known or optimal solutions in the literature and shown to be very effective and efficient in finding solutions to 2

Computers & Industrial Engineering 139 (2020) 106109

K. Karabulut and M.F. Tasgetiren

the existing problem instances. The proposed ES approach obtained new best-known solutions for 7 problem instances for the first time in the literature. The rest of the paper is organized as follows: Section 2 presents the KT heuristic, Section 3 gives details of the ES algorithm for the TOPTW, and the computational experiments with parameter setting are given in Section 4. Finally, the conclusion is given in Section 5.

The above measure is used in two insertion heuristics in Labadie et al. (2011). The first heuristic, denoted as LI1, starts from the first tour and considers the insertion of all unvisited nodes only to the last position of the current tour. The node with the highest qj P j / j ratio is selected for insertion. If there is no feasible insertion to the current tour, the next tour is considered. This process continues until m tours are constructed. In the second heuristic, denoted as LI2, all nodes are sorted according to their profits in a decreasing order and m nodes with the highest profits are inserted into m tours. Then, all unvisited nodes are considered for insertion into all positions in all tours. Again, the insertion position and the unvisited node that have the highest ratio are selected. Another heuristic, denoted as LI3 is proposed by Labadie et al. (2011). LI3 starts with sorting all nodes according to their profits in a decreasing order and considers the node at the top of the list for insertion into all positions in all tours. The insertion with a minimum j is selected. If there is no feasible insertion for the current node, the next node in the unvisited list is considered. The heuristic stops when all nodes are considered for insertion. In addition to the above, two more heuristics (LS1 and LS2) are also proposed by Labadie et al. (2011). They are adaptions of the well-known sweep heuristic for vehicle routing problems. Profits and increases in travel costs for visited nodes should be carefully considered for constructing good solutions and have been considered in heuristics proposed by Vansteenwegen et al. (2009a, 2009b, 2009c) and Labadie et al. (2011). Another important point is the latest visiting time. The nodes with sooner latest visiting times would be harder to add to the partial solutions later. The constructive heuristic proposed in this study takes the latest visiting time of a candidate node into account, in addition to the profit and cost increase associated with the node. The proposed heuristic uses a ratio calculated for each feasible insertion as follows:

2. The constructive heuristic Constructive heuristics aim to develop good solutions/individuals by starting from an empty solution and iteratively adding solution components to the current partial solution. Since the TOPTW is a constrained optimization problem, a constructive heuristic must ensure that the solution is feasible when new components are added to the partial solution. Constructive heuristics generally use a problem-specific cost function to select the next candidate node to insert into the partial solution. One such function is given by Vansteenwegen et al. (2009a, 2009b, 2009c). Three insertion heuristics and two heuristics based on sweep algorithm to construct initial solutions are developed by Labadie et al. (2011). The insertion move tries to add one more node to the current solution. When a new node is considered for insertion into one of the current tours, it should be checked whether the inserted node and the nodes after the insertion point have their services started before their latest service starting time, in order to ensure that the solution remains feasible. If not, the insertion of the node will be infeasible. When making insertions, a ranking method based on the cost function is used to select the next node from the unvisited list of nodes. Insertion of a new node may increase both travel costs and profits or vice versa. A straightforward approach is to select a node that will add more profit with less travel cost. Since insertion neighborhood typically makes insertion of all unvisited nodes in all positions of all tours of a solution, it consumes substantial computation time. If the insertion of a node into a particular position in a particular tour is feasible, the imposed increase in the total travel cost can be calculated faster as explained by Vansteenwegen et al. (2009a, 2009b, 2009c) as follows: The speed-up method for feasibility check for insertion can be implemented by keeping track of waiting times (wi ) and maximum delay time for the start of service (di ) for all visited nodes. wi is the waiting time at node i before the service starts. It is positive if the arrival time is before the earliest visiting time or 0 otherwise: wi = max(ei Ai , 0) . di is the maximum possible delay time for node i , which is either limited by its own latest visiting time or by the next node’s in the permutation, i.e., di = min(li SSTi , wi + 1 + di + 1) . The travel time increase when an unvisited node j is inserted between nodes i and i + 1 can be calculated as: j

= ci, j + wj + Sj + cj, i + 1

ci, i + 1

e

wi + 1 + di + 1

P j2/

j

(6)

where lmax is the maximum latest visiting time. The heuristic uses the ratio of the squared additional profit divided by delta travel time increase developed by Vansteenwegen et al. (2009a, 2009b, 2009c). Since the problem includes time windows, the latest visiting time of the possible new node to be inserted is also important. For two alternative insertions with equal or close P j2/ j ratios, the node with sooner latest visiting time should be chosen so that other feasible insertions could be possible in later iterations of the heuristic. The proposed ordering scheme favors the nodes with sooner latest visiting times and insertions with the smaller cost for the same node. Note that throughout the paper, we use this desirability measure in all parts of the ES algorithm wherever necessary when making feasible insertions. The computational experiments presented in Section 4 show that the proposed KT heuristic generates slightly better results on overall average when compared to the other six heuristics developed by Vansteenwegen et al. (2009a, 2009b, 2009c) and Labadie et al. (2011).

(4)

For a feasible insertion, j should be less than or equal to the sum of waiting time and maximum possible delay for node i + 1: j

(l j / lmax )

(5)

3. Evolution strategy approach

A “desirability measure” can be calculated for each feasible insertion. Vansteenwegen et al. (2009a, 2009b, 2009c) employed an insertion method that all nodes are inserted in all tours, and chose the node with the highest P j2/ j ratio. They state that travel time increase becomes less important than profit because of the time windows, therefore profit is squared. They also report that worse results are obtained when profit is not squared. We denote this heuristic as VI insertion heuristic. Another measure is presented in Labadie et al. (2011), which is qj P j / j , where qj is the number of reachable neighbors (these are the neighbors that can be visited without a constraint violation) from node j . A candidate node with more reachable neighbors will have a higher priority, assuming that this scheme will increase the probability of finding other feasible insertions in the next iterations.

Evolution strategy (ES) is a metaheuristic optimization algorithm developed by Rechenberg (1971) and Schwefel (1975). ES is typically applied to real parameter optimization problems and it uses mutation for offspring generation. The mutation is performed by adding a random value drawn from a normal distribution. Parameter tuning is an important issue when using evolutionary algorithms, including ES. The idea of evolving parameters within the individuals, self-adaption, is considered as a standard approach in ES (Beyer & Schwefel, 2002). The main assumption in evolving parameters within individuals is; a good parameter value that generates a good individual survives together with that individual. 3

Computers & Industrial Engineering 139 (2020) 106109

K. Karabulut and M.F. Tasgetiren

0

5

1

4

2

3

0

0

5

3

0

2

1

0

Fig. 3. An example of N2.

which a number of nodes are removed from the current solution and added to the solution again in a way that it optimizes the chosen criterion. It is shown to generate very good offspring solutions for combinatorial optimization problems by Schrimpf et al. (2000). The single and very important parameter to be adjusted for the RR algorithm is the ruin size, which is the number of nodes to be removed from the solution. A small value does not create a move that is large enough to escape local optima, while a large value destroys a good solution. As stated before, a self-adaptive scheme can be used instead of presetting such a strategy parameter. The ruin size for the offspring is calculated by mutating the current ruin size as rSi = rSi. e N (0,1) , where is the learning rate, N (0, 1) is a random number generated from a unit normal distribution with mean zero and standard deviation one. The ruin size is mutated by the unit normal distribution, N (0, 1) , that leads to a small variation with a high probability and a large variation with a low probability, in order to obtain the offspring ruin size. If the offspring ruin size is outside the predetermined limits, it is restricted to the predetermined levels. In the ruin part of the RR algorithm, a number rSi of nodes are removed from the tours as follows: For each node to be removed, a random tour is selected from individual i , then, a single node is removed from the tour and the node is added to the list U of unvisited nodes. This is repeated for rSi times. In the recreate part, we use the neighborhood structure, N0 = MoveFromUnvisitedList , which is inspired from the multiple insertion heuristic (MIH) proposed for solving parallel machine scheduling problem with sequence-dependent setup times in Kurz and Askin (2001). This neighborhood structure aims at increasing the profits of tours, hence the fitness of a solution. Simply, each node in the unvisited list U is inserted into all positions in all tours in the hope of increasing the profits with an additional feasible node. Note that the neighborhood structure N0 is computationally very expensive because of inserting a node to every possible position in m tours. In order to accelerate it, we take advantage of the nearest neighbor approach for insertions in Snyder and Daskin (2006). Suppose that there is a tour Tk with total cost TCk and node x will be inserted into an edge {i , j} . The cost of the new tour can be easily recalculated by TCknew = TCk + ci, x + cx, j ci, j . It can be used in all tours so that insertions are substantially accelerated in order to compute the total travel cost, TTC. The best feasible insertion is determined by using the proposed metric in Eqs. (5) and (6) while TTC is determined using the nearest neighbor approach. Note that for insertion of node x to first and last position, the nearest neighbor approach cannot be used but it can be easily modified as given in Tasgetiren, Suganthan, and Pan (2007), (2010). The recreate procedure stops when the last node in U is inserted into all positions in all tours. To illustrate, we consider the example in Fig. 1. Suppose that rSi = 2 . In the ruin phase, two nodes will be removed from the individual i . Suppose we select node 4 from the first tour and node 7

Fig. 1. Solution representation.

3.1. Solution representation Solution representation for the self-adaptive ES uses one permutation for each tour and one list for the unvisited nodes and the ruin size parameter value rS that is the number of nodes that will be removed from the solution in the ruin and recreate procedure. The details of the ruin and recreate procedure will be presented later. In the solution representation, each solution employs m partial permutations in order to represent m tours. Each tour starts and ends at the depot, which is node 0. However, the depot is not included in the permutations. Let Tk be the k th tour, then, a solution can be represented by Tk (k = 1, ..,m) . Nodes that are not present in any of the tours are stored in U , which is the list of unvisited nodes. Note that cij is the cost of the distance between nodes i and j . Let TCk be the cost of the tour Tk . In the solution representation, we also keep the objective function value (fitness) of all m tours as f ( ) , which is the total collected profit for all m tours, as well m as the total travel cost of the solution, TTC = k = 1 Tk . For example, given N = {1, 2, ..,10} and m = 2, Fig. 1 illustrates the solution representation: 3.2. Initial population Three individuals in the initial population with size | | are constructed using three of the constructive heuristics: LS1, VI and KT. We use only these heuristics, not the other four, because these three produce better results and take less CPU time when compared to the other heuristics. For the rest of the initial population, one of these three individuals are selected randomly, and is perturbed by applying two neighborhood structures randomly selected out of four neighborhood structures used in the local search procedure that is explained in section 3.4. The local search procedure given in Algorithm 1 is also applied to all individuals in the initial population and the best so far individual is recorded. The evolution loop is repeated until the predefined time limit or an optimum solution is reached. The number of offspring generated in each iteration is seven times the population size (Eiben & Smith, 2003). 3.3. Offspring population Parent selection for reproduction is done by using a discrete uniform distribution, DU (1, | |) , thus, each individual has an equal chance for selection. Once an individual i from the parent population | | is selected randomly, the offspring individual µi is generated by using the ruin and recreate (RR) algorithm (Schrimpf, Schneider, StammWilbrandt, & Dueck, 2000). RR is a simple but effective method, in

0

3

1

0

2

0

0

0

3

0

0

1

2

4

Fig. 2. An example of N1.

0

Computers & Industrial Engineering 139 (2020) 106109

K. Karabulut and M.F. Tasgetiren

Fig. 4. An example of N3.

0

3

1

0

0

3

2

0

0

2

4

0

0

1

4

0

0

3

1

0

3

4

0

2

0

0

2

0

Unvisited list

4

0

Unvisited list

5

0

1

5

Fig. 5. An example of N4 . Table 1 Average gaps and running times of constructive heuristics. LI1

LI2

LI3

LS1

LS2

VI

KT

Instance

Gap (%)

t (ms)

Gap (%)

t (ms)

Gap (%)

t (ms)

Gap (%)

t (ms)

Gap (%)

t (ms)

Gap (%)

t (ms)

Gap (%)

t (ms)

C (m = 1) C (m = 2) C (m = 3) C (m = 4) S (m = 1) S (m = 2) S (m = 3) S (m = 4) V (S) V (C) Average

27 27 25 20 18 16 13 10 7 7 17.07

3.49 4.76 7.74 9.27 0.84 1.2 1.38 1.47 1.68 12.99 4.48

25 25 21 20 11 10 7 7 5 8 13.87

10.86 30.62 54.87 85.67 3.73 9.18 13.35 16.68 23.68 327.9 57.65

46 40 36 32 22 19 15 12 8 12 24.19

0.07 0.09 0.13 0.16 0.04 0.07 0.09 0.1 0.13 0.4 0.13

24 24 22 20 11 11 10 6 7 12 14.63

1.12 2.15 2.43 2.27 0.74 0.66 0.43 0.35 0.35 0.45 1.09

38 35 32 29 37 29 23 19 12 15 26.89

0.88 0.73 0.65 0.58 0.52 0.33 0.22 0.17 0.22 0.24 0.46

19 20 16 14 8 9 7 6 5 5 10.88

0.68 2.08 4.05 6.32 0.45 1.09 1.55 1.84 3.04 25.96 4.71

21 19 16 13 8 9 7 6 4 6 10.86

0.78 2.33 4.5 7 0.53 1.25 1.82 2.23 3.79 30.08 5.43

C: Cordeau; S: Solomon; V: Vansteenwegen.

of both T1and T2 . Suppose that there is no increase in total profit with a feasible insertion from them. Then, node 6 is tried. Suppose that there is no increase in total profit with a feasible insertion from them, again. Then, node 9 is tried. Suppose that there is an increase in total profit with a feasible insertion in position 2 of tour T2 . Next, node 10 is tried. Suppose that there is an increase in total profit with a feasible insertion in position 3 of tour T1. Now, nodes 4 and 7 are also tried. Suppose that there is no increase in total profit with a feasible insertion from them. Ultimately, we end up with a new offspring with feasible tours and increased profit as follows:

Table 2 ANOVA results for parameters of ES.

µ *µ Residuals

Sum Sq

Df

F value

Pr(> F)

0.000033714 0.000003700 0.000028355 0.000182893

3 3 9 64

3.9325 0.4316 1.1025

0.01222 0.73109 0.37397

from the second tour. Then, nodes 4 and 7 will be removed from added to the end of the unvisited list U as follows:

i

and

T1

{2, 8, 10}

T1

{2, 8}

T2

{1, 9, 5}

T2

{1, 5}

U

{3, 6, 4, 7}

U

{3, 6, 9, 10, 4, 7}

TC1

c0,2 + c2,8 + c8,10 + c10,0

TC2

c0,1 + c1,9 + c9,5 + c5,0

TC1 TC2

c02 + c28 + c80 c01 + c15 + c50

TTC

TC1 + TC2

f (µ )

P2 + P8 + P1 + P5

TTC

TC1 + TC2

f (µ )

P2 + P8 + P10 + P1 + P9 + P5

3.4. Local search

Then, in the recreate phase, we employ the neighborhood structure, N0 , in order to insert all nodes in U into all positions in T1 and T2 in order to find a new feasible offspring with an increased total profit. In other words, first, node 3 is removed from U , and inserted into three positions

We employ a random local search consisting of four different neighborhood structures. Let Nk be the k th neighborhood structure. The 5

Computers & Industrial Engineering 139 (2020) 106109

K. Karabulut and M.F. Tasgetiren

• •

Fig. 6. Plot of

Briefly, the local search begins with applying relocation moves, which is applied to each tour separately. They try to relocate nodes within each tour in order to decrease the travel cost for that tour. The backward relocation (BR) procedure takes nodes starting from the last node in the current tour and tries to move the nodes so that they are visited earlier. The forward relocation (FR) procedure starts from the first visited node in the current tour and tries to move the nodes so that they are visited later. Feasibility check and change in travel cost are computed by using the previously mentioned speed-up methods described in Section 2, and the nearest neighbor approach in Section 3.3, respectively. Relocation procedures use first improvement pivoting rule, i.e., the node is relocated as soon as a relocation that lowers the travel cost is found. If the travel cost is decreased by the relocation moves, it is checked whether one or more unvisited nodes can be added to the current solution by using the neighborhood structure, N0 . Next, a random local search based improvement procedure is executed. One neighborhood

and µ against the average gap/RD.

following four neighborhood structures are proposed for the local search:

•N

•

random tour and then swaps two randomly chosen nodes in that tour. Fig. 3 provides an example where nodes one and three in the tour are swapped. N3 = SwapNodesBetweenTours : This neighborhood structure chooses two random tours and two random nodes in those tours and swaps them. Fig. 4 provides an example where node one in tour one is swapped with node two in tour two. Similar to N1, this neighborhood is not used when there is only one tour (m = 1). N4 = SwapNodeWithUnvisited : This neighborhood aims at achieving one of the following goals by swapping a visited node with an unvisited node. It either increases profit by swapping a visited node with an unvisited one that has a higher profit, or decreases the total travel cost for the corresponding tour if the profits of the visited and unvisited nodes are the same or both if the swap results in profit increase and travel cost decrease at the same time. Fig. 5 provides an example where node one in tour one is swapped with unvisited node four, making node four visited in tour one and node one unvisited.

1 = MoveNodeToAnotherTour : This neighborhood structure chooses a random node from a random tour and moves it to another randomly chosen tour in a random position in that tour. Fig. 2 provides an example where node one in tour one is moved to position one in tour two. Note that this neighborhood is not used when there is only one tour (m = 1). N2 = SwapNodesWithinTour : This neighborhood structure selects a

Table 3 Overall comparison of ES with existing algorithms on the instances of Cordeau and Solomon. ES

CP

I3CH

GVNS

SSA

ILS

ABC

GRILS

ILSI3CH

SAILSI3CH

ILSVNS

SAILSVNS

AG

AT

AG

AG

AT

AG

AT

AG

AT

AG

AT

AG

AT

AG

AT

AG

AG

AT

AG

AG

AT

m=1 C 1–10 C 11–20 S 100 S 200 Avg.

0.74 3.35 0.02 0.69 0.79

40.7 33.7 0.9 35.8 22.9

1.35 3.81 0.47 3.55 2.12

1.05 4.31 0.69 1.58 1.53

109.0 130.2 26.7 132.1 88.6

0.54 3.21 1.40 2.15 1.79

12.4 24.2 66.5 75.5 57.0

0.97 3.75 0.05 1.09 1.03

112.2 162.4 22.3 44.7 60.5

4.72 9.59 1.94 3.11 3.73

1.8 2.0 0.2 1.7 1.2

1.47 3.56 0.46 1.56 1.39

78.5 103.7 4.4 21.4 33.2

3.73 7.99 0.27 2.65 2.59

4.2 5.2 1.4 8.4 4.8

0.66 2.28 0.00 0.88 0.70

0.24 1.83 0.00 0.32 0.39

126.8 151.5 31.1 153.7 103.1

0.57 1.70 0.00 0.47 0.47

0.66 1.84 0.03 0.52 0.53

753.6 958.9 78.3 786.4 534.6

m=2 C 1–10 C 11–20 S 100 S 200 Avg.

1.35 2.21 0.22 0.72 0.81

55.5 76.1 16.7 80.8 52.4

3.38 5.95 1.24 1.44 2.21

1.20 3.07 0.50 0.64 0.98

247.1 304.6 69.3 463.8 263.8

0.91 1.57 0.87 0.99 1.01

39.1 82.4 73.9 19.8 51.2

2.54 4.24 0.16 1.12 1.35

173.9 201.6 34.5 76.9 89.9

6.31 8.19 1.96 3.26 3.82

4.8 5.2 0.9 2.6 2.6

3.13 4.00 0.44 1.32 1.58

149.7 221.8 13.0 30.6 64.7

6.85 11.61 1.75 3.91 4.48

11.7 13.5 4.0 13.7 9.7

2.32 2.78 0.12 1.18 1.14

0.87 1.83 0.06 0.55 0.57

287.4 354.3 80.6 539.5 306.9

1.90 2.73 0.01 1.12 1.01

1.43 2.38 0.12 0.90 0.87

481.1 567.2 63.1 746.0 427.0

m=3 C 1–10 C 11–20 S 100 S 200 Avg.

0.99 1.97 0.32 0.10 0.55

95.9 123.7 21.5 17.5 43.3

4.70 5.67 1.95 0.04 2.12

0.43 1.24 0.21 0.02 0.31

424.0 497.0 135.9 89.2 204.7

0.43 1.15 0.90 0.14 0.60

85.9 150.7 91.1 7.3 68.5

2.40 3.94 0.45 0.48 1.18

197.0 251.8 45.9 52.3 95.2

6.64 9.31 2.41 1.13 3.42

9.2 9.7 1.5 1.7 3.7

2.97 4.36 0.64 0.35 1.33

228.9 247.6 22.5 12.5 75.7

7.64 11.33 3.20 1.65 4.30

18.4 20.3 6.4 14.1 12.5

2.79 3.21 0.45 1.65 1.55

1.46 1.79 0.25 0.66 0.76

493.2 578.1 158.0 103.8 238.1

2.78 3.20 0.28 0.33 1.01

2.26 2.49 0.63 0.33 0.98

433.8 474.5 63.1 295.5 248.6

m=4 C 1–10 C 11–20 S 100 S 200 Avg. Grand Avg.

1.42 2.23 0.55 0.00 0.69 0.71

137.7 160.3 56.6 0.0 60.8 44.9

5.41 6.27 2.38 0.00 2.44 2.22

0.67 0.53 0.26 0.00 0.26 0.77

566.5 728.6 199.5 0.2 246.6 200.9

1.38 2.13 1.18 0.00 0.91 1.08

127.3 232.6 86.6 0.5 80.6 64.3

2.53 4.03 0.67 0.00 1.12 1.17

255.6 284.0 58.3 40.4 107.6 88.3

7.37 8.54 3.30 0.00 3.35 3.58

14.1 13.7 2.4 1.0 4.9 3.1

3.78 3.91 1.14 0.00 1.45 1.44

233.3 267.6 29.5 0.5 77.4 62.8

8.37 10.41 5.15 0.00 4.44 3.95

25.4 27.2 8.9 14.3 15.4 10.6

3.48 3.53 1.11 0.18 1.41 1.20

1.98 1.89 0.48 0.06 0.71 0.61

659.0 847.6 232.1 0.2 286.9 233.7

3.61 3.88 0.85 0.00 1.31 0.95

3.30 3.24 1.27 0.00 1.35 0.93

369.6 374.1 61.2 129.4 167.2 344.3

AG = Aggregated gap for instances: AT = Aggregated CPU time for instances. 6

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K. Karabulut and M.F. Tasgetiren

Table 4 Overall comparison of ES with existing algorithms on the instances of Vansteenwegen. ES

C 1–10 S 100 S 200 Average

CP

I3CH

GVNS

SSA

ILS

ABC

ILS100

ILS35

ILS10

SAILS100

SAILS35

SAILS10

AG

AT

AG

AG

AT

AG

AT

AG

AT

AG

AT

AG

AT

AG

AG

AG

AG

AG

AG

0.96 0.15 0.02 0.22

78.3 60.5 12.8 43.7

0.89 0.24 0.07 0.27

0.78 0.03 0.04 0.15

326.7 393.8 127.2 274.5

1.25 1.14 0.12 0.74

51.3 29.7 3.2 22.1

1.04 0.59 0.08 0.45

566.0 90.8 48.4 145.4

2.32 1.80 0.39 1.30

30.4 3.2 1.5 6.6

1.05 0.53 0.07 0.42

181.1 41.2 5.9 48.0

0.96 0.47 0.02 0.36

1.11 0.18 1.68 0.93

1.23 0.19 1.84 1.02

0.92 0.39 0.09 0.35

1.05 0.12 1.53 0.84

1.05 0.12 1.43 0.80

Interval Plot of Algorithms 95% CI for the Mean 8 7 6

Gap

5 4 3 2 1 0 ES

CP

CH I3

NS GV

A SS

S IL

AB

C

I GR

LS IL

H 3C SI

IL SA

H 3C SI

IL

NS SV

I SA

S VN LS

Fig. 7. Interval plot of the mean Gaps for Cordeau and Solomon instances.

Algorithm 1. (LS(g, TTC(g))).

Table 5 New best solutions found by ES. Instance

m

BKS

New BKS

r209 pr13 pr15 pr10 pr03 pr08 pr20

2 2 2 3 4 4 4

1414 832 1220 1573 1246 1385 2062

1417 845 1222 1574 1247 1392 2082

g' g'

if (TTC (g ') < TTC (g )) then g Itermax n Iter 1 do while (Iter Itermax )

CPU

CPU Mark

Multiplier

ES I3CH GVNS ILS, SSA ABC ILSX, SAILSX GRILS

Intel Core2 Quad Q9400 @ 2.66 GHz Intel Xeon E5430 @ 2.66 GHz Intel Pentium 4 3.06 GHz Intel Core2 Duo T9300 @ 2.50 GHz AMD Athlon X2 250 Intel Core i5-650 @ 3.20 GHz Intel Xeon L5420 @ 2.50 GHz

1128 1133 657 978 991 1323 1046

1.72 1.72 1.00 1.49 1.51 2.01 1.59

N0 (g ')

g' g k DU (1, 4) if (k = 1) then g ' N1 (g ') else if (k = 2) then g ' N2 (g') else if (k = 3) then g ' N3 (g ') else g ' N4 (g') if (f (g') > f (g ) or (TTC (g ') < TTC (g )) then g N0 (g ') Iter 1 else Iter Iter + 1 endif enddo endAlgorithm

Table 6 CPU Mark results of different algorithms. Algorithm

BR (g ) FR (g ')

amongst four possible neighborhoods is chosen randomly. If the chosen neighborhood results in a decreased travel cost or a better fitness value, then, N0 is executed again and the iteration counter is reset to one. Otherwise, the iteration counter is incremented by one. The local search continues until a better solution cannot be found in n iterations. The pseudo-code of the local search is given in Algorithm 1.

3.5. Selection After having generated offspring population |µ|, they are added to µ ) will the parent population | |, the combined population Q = ( have to be truncated to its initial size before the next iteration. The truncation is done by calculating a goodness score proposed by Lü, Glover, and Hao (2010) and sorting the individuals according to their 7

Computers & Industrial Engineering 139 (2020) 106109

K. Karabulut and M.F. Tasgetiren

Fig. 8. Scatter plot of the mean gaps vs adjusted CPU times for Solomon instances.

Fig. 9. Scatter plot of the mean gaps vs adjusted CPU times for Cordeau instances.

scores. The individuals with the highest goodness scores are selected for the next generation and the others are removed from the population. The goodness score for an individual is calculated as GSi = N (f (Qi)) + (1 ) N (disti ) where is a weighting parameter, N (Qi ) is a normalization function, f (Qi) is the fitness value of individual Qi and disti is the distance of individual Qi to the population. The normalization function is defined as N (Qi ) = (f (Qi ) f (Qmin))/(f (Qmax ) f (Qmin) + 1) , where f (Qmin) is the minimum and f (Qmax ) is the maximum f (Qi) value in the population. The distance of individual Qi to the population is its distance to the most similar individual in the population and is calculated as disti = min(distij, j i ) where distij is the distance between individuals Qi and Qj . Consequently, distij is the number of different nodes visited in

individual Qi . distij can be calculated in shorter time by counting the number of common nodes (nSize ) in the unvisited list U and subtracting this number from the total number of visited nodes in all routes, instead of comparing all visited nodes in all routes in two individuals and counting the number of common visited nodes: distij = n nSize . The pseudo code of the proposed ES is given in Algorithm 2. Algorithm 2. (Evolution strategy). Construct initial population Apply local search and find best while (Not TerminationCriterion) for i 1 to 7 | |

8

Computers & Industrial Engineering 139 (2020) 106109

K. Karabulut and M.F. Tasgetiren

Fig. 10. Scatter plot of the mean gaps vs adjusted CPU times for Vansteenwegen problems. Table 7 Analysis of Local Search Components. ES

No Local Search

Only RL

Only RL + N0

Only RL + N1

Only RL + N2

Only RL + N3

Only RL + N4

% Gap

time (s)

% Gap

time (s)

% Gap

time (s)

% Gap

time (s)

% Gap

time (s)

% Gap

time (s)

% Gap

time (s)

% Gap

time (s)

S C S C S C S C S C

0.35 2.04 0.47 1.78 0.21 1.48 0.28 1.82 0.09 0.96

17.7 37.2 47.6 65.8 19.6 109.8 29.3 149.0 37.5 78.3

0.51 2.39 0.48 1.86 0.21 1.73 0.35 1.99 0.08 0.91

22.7 28.8 43.0 76.9 22.1 105.1 43.0 76.9 56.2 156.3

0.53 2.11 0.47 1.99 0.21 1.84 0.34 1.82 0.09 0.93

22.9 34.9 47.8 69.7 26.1 108.5 26.4 138.8 71.5 127.8

0.54 2.11 0.47 1.99 0.21 1.84 0.34 1.82 0.22 0.19

21.2 35.1 47.7 69.5 26.1 108.3 26.3 136.6 63.2 167.9

n/a n/a 0.56 1.89 0.26 1.69 0.36 1.68 0.08 0.94

n/a n/a 51.0 83.0 24.3 99.3 26.1 141.4 55.9 141.3

0.53 2.11 0.41 1.79 0.19 1.59 0.35 1.82 0.08 0.94

21.8 40.4 40.9 66.4 21.3 105.9 32.0 137.9 61.4 115.3

n/a n/a 0.51 1.92 0.23 1.54 0.33 1.75 0.08 0.99

n/a n/a 45.3 87.1 20.1 104.3 24.9 123.7 60.1 129.1

0.54 2.11 0.52 1.81 0.25 1.66 0.34 1.87 0.09 0.94

19.9 37.8 48.6 59.1 19.4 120.7 27.1 128.1 69.2 139.1

S C

0.28 1.62 0.95

30.4 88.0 59.2

0.33 1.78 1.05

37.4 88.8 63.1

0.33 1.74 1.03

39.0 95.9 67.4

0.36 1.59 0.97

36.9 103.5 70.2

0.31 1.55 0.93

39.3 116.2 77.8

0.31 1.65 0.98

35.5 93.2 64.3

0.29 1.55 0.92

37.6 111.1 74.3

0.35 1.68 1.01

36.8 96.9 66.9

Avg without m = 1 S 0.26 C 1.51

33.5 100.7

0.28 1.62

41.1 103.8

0.28 1.64

43.0 111.2

0.31 1.46

40.8 120.6

0.31 1.55

39.3 116.2

0.26 1.54

38.9 106.4

0.29 1.55

37.6 111.1

0.30 1.57

41.1 111.7

Avg

67.1

0.95

72.5

0.96

77.1

0.89

80.7

0.93

77.8

0.90

72.6

0.92

74.3

0.94

76.4

m=1 m=2 m=3 m=4 Opt Avg Avg

0.89

RL = Backward and forward relocate. µi

return best end Algorithm

DU (1, | |)

rSi rSi. e N (0,1) if (rSi < rSmin) then rSi rSmin rSmax number of visited cities in µi if (rSi > rSmax ) then rSi µi Ruin (µi ) µi

No (µi )

µi

LS (µi )

if (f (µi ) > f ( µi best

rSmax

4. Computational experiments The proposed constructive heuristic and evolutionary strategy are coded in C++ and experiments are carried out on a computer having a 2.66 GHz Intel Core2 Quad Q9400 CPU and 3GBs of main memory. Two benchmark sets are used in experiments. There are 144 problem instances in total in these two benchmark sets. The first set is designed by Montemanni and Gambardella (2009) based on their OPTW instances, which in turn are based on Solomon (1987) vehicle routing with time windows test instances and Cordeau, Gendreau, and Laporte (1997) multi-depot vehicle routing problems. These problems have up to four tours. Some of these problem instances have been solved to optimality.

best ))

else if (TTC (µi ) < TTC ( µi best

best )

and f (µi )=f (

best ))

then

endif Add i to population Q endfor Calculate goodness scores (GS ) for all individuals in Q = (µ ) Sort population Q by GS in a decending order Keep | | best individuals from Q for the next generation and remove others endwhile 9

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K. Karabulut and M.F. Tasgetiren

The second benchmark set was designed by Vansteenwegen et al. (2009a, 2009b, 2009c) based on the original Solomon (1987) and Cordeau et al. (1997) instances and use the same number of tours as vehicles, so that, it is possible to visit all nodes. In this way, optimal solutions are equal to the sum of all profits.

Gunawan et al. (2017), which can be found in Appendix A from Table A1 to A10. We compare the results of the ES to the updated best-known (BK) or optimal (Opt) solutions in the Appendix A. It is important to note that average gaps and CPU times of the ES are used in all comparisons in all tables in Appendix A since ES is a stochastic algorithm. In addition, we also provided the best, average, worst solution values as well as average CPU times. We compare the performance of the ES with CP, I3CH, GVNS, SSA, ILS, ABC, GRILS, ILSI3CH, SAILSI3CH, ILSVNS and SAILSVNS for Cordeau, Solomon, and Vansteenwegen problems. In addition, we compare the ES with CP, I3CH, GVNS, SSA, ILS, ABC, ILS100, ILS35, ILS10, SAILS100, SAILS35 and SAILS10 for Vansteenwegen problems. Note that CP is an exact method, whereas ILS and I3CH are deterministic algorithms with only a single run. Only the best solutions values are reported for the SSA algorithm. Finally, the ABC, GRILS, ILSI3CH, SAILSI3CH, ILSVNS, and SAILSVNS are stochastic algorithms. For these reasons, average values of ES, GVNS, ABC, GRILS, and different CPU variants of ILSX and SAILSX are compared to the best values of ILS, I3CH and SSA algorithms. Even though the detailed results are given in Table A1–A10, in Appendix A for ES, CP, I3CH, GVNS, SSA, ILS, ABC algorithms, the detailed results for GRILS, ILSX and SAILSX algorithms are not provided in the literature. For this reason, we directly take the aggregated values presented in their papers for these three algorithms in the following analyses. Since it is difficult to make an analysis of competing algorithms due to the complexity of computational results, we provide Table 3 and Table 4, where the results are aggregated as in Gunawan et al. (2017). Note that we only evaluate the algorithms in terms of solution quality below. We analyze the performance of algorithms on instances of Cordeau and Solomon. From Table 3, it can be observed that the best average percent gaps (AG values) for m = 1 are obtained by SAILSI3CH, ILSVNS, SAILSVNS are 0.39%, 0.47%, and 0.53%, respectively. However, ILSI3CH and ES algorithms are very competitive with them having gaps with 0.70% and 0.79%, respectively. For m = 2, the best result with a gap of %0.57 is produced by SAILSI3CH algorithm while ES and SAILSVNS are also very competitive with it by generating gaps with 0.81% and 0.87%, respectively. Regarding m = 3, the best algorithm is I3CH with a gap of 0.31% followed by ES and GVNS with 0.55% and 0.60% gaps, respectively. Finally, for m = 4, again, the best algorithm is I3CH with a gap of 0.26% followed by ES and SAILSI3CH with 0.69% and 0.71% gaps, respectively. However, one can have a look at the grand average of 16 instances. It can be observed from the grand average in Table 3 that the best algorithms are SAILSI3CH, ES and I3CH by generating gaps of 0.61%, 0.71%, and 0.77%, respectively. Note that SAILSI3CH and I3CH algorithms are computationally very expensive ones when compared to ES that will be analyzed by adjusting the CPU times for different machines used by algorithms. Now, we analyze the performance of algorithms on instances of Vansteenwegen. From Table 4, it can be observed that ILSX and SAILSX algorithms are not so much competitive to I3CH, ES and CP algorithms. The best algorithm is I3CH generating a gap of 0.15% while ES and CP are able to generate very competitive results with gaps of 0.22% and 0.27%, respectively. Table 3 Overall comparison of ES with existing algorithms on the instances of Cordeau and Solomon In order to picture out if the differences in Gaps are significant in Table 3, we provide the interval plot of the algorithms in terms of AG (average Gap) values in Fig. 7. Note that, for confidence interval plots, if the confidence interval of algorithm A does not overlap with the confidence interval of algorithm B, differences in mean gaps between A and B are statistically significant and meaningful. As seen in Fig. 7, the differences in Gaps between SAILSI3CH, I3CH, ES and CP, ILS, GRILS are significant; hence, meaningful since their confidence intervals do not overlap. In terms of mean Gap values, the best algorithms are SAILSI3CH, I3CH and ES algorithms. However, SAILSI3CH, ILSVNS; SAILSVNS, ILSI3CH, GVNS, I3CH and ES algorithms are equivalent because their

4.1. Computational results for constructive heuristics The KT heuristic proposed in this paper, as well as the heuristics used in Labadie et al. (2011) and Vansteenwegen et al. (2009a, 2009b, 2009c), are also coded and executed on the same machine. In order to compare the performance of the heuristics, the gap is used as a metric which is the difference between the fitness values of the solutions generated by the corresponding heuristic to the best-known or optimum fitness values. The average gap and the average time required to find the solutions by each heuristic for each problem set are given in Table 1. From Table 1, it can be seen that the proposed KT heuristic and VI heuristic proposed by Vansteenwegen et al. (2009a, 2009b, 2009c) are superior to all the other heuristics since gaps of 10.86 by KT and 10.88 by VI heuristics are smaller than LI1, LI2, LI3, LS1, and LS2 heuristics. The KT heuristic produces slightly better results than the VI heuristic. LI2 performs better than the other four heuristics, however, its running time is very high when compared to the other heuristics. 4.2. Parameter tuning The proposed ES has two parameters to tune. They are the population size µ and value used in goodness score calculation. We carried out a full factorial design by using design of experiments (DOE) approach (Montgomery, 2012) and we took the following values for the parameters: µ = {15, 20, 25, 30} and = {0.7, 0.75, 0.8, 0.85} . We followed the scenario used by Hu and Lim (2014) and Gunawan et al. (2017) to determine the test instances. The chosen instances are c203, c207, pr02, pr07, pr12, pr16, r102, r105, rc107 and rc204 with m equals to four. In the design of ES, there are 4 × 4 = 16 algorithm configurations, i.e. treatments. Each instance is run for 5 times for 16 treatments with a maximum CPU time being equal to Tmax = 20 seconds. We calculate the relative deviation (RD/Gap) for each instance-treatment pair as follows:

RPD =

Fbest F FBest

100

(7)

where Fi is the total profit generated by the ES in each treatment, and Fmin is the minimum total profit found amongst 16 treatments. For each instance-treatment pair, the average RD value is calculated by taking the average of 10 instances in each treatment. Then, the response variable average (RD) of each treatment is obtained by averaging these RD values of all treatments. ANOVA results are given in Table 2. Two-way ANOVA results showed that parameter was significant with a p-value of 0.01222 that is less than = 0.05, while µ and : µ interaction is not significant with p values of 0.73109 and 0.37397, respectively. However, since was significant, we examine the interaction plot, which is given in Fig. 6, Fig. 6 indicates that the values of µ = 15 and = 0.7 produced the lowest RD/gap. 4.3. Computational results for evolution strategy The maximum CPU time is limited to 100 s for Solomon instances and 300 s for Cordeau instances. The ES is executed five times on each problem instance. Parameters for the ES are set as follows: µ = 15; = 1; rSmin = 2 ; and = 0.7 . It should be noted that the best-known or optimal solutions so far in the literature are carefully updated by considering the best solutions from ES, CP, I3CH, GVNS, SSA, ILS, and ABC as well as from 50 new best known solutions presented very recently in 10

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K. Karabulut and M.F. Tasgetiren

confidence intervals do overlap. To sum up, ES is very competitive to I3CH and ILSX and SAILSX variants in terms of gap values as seen in analyses above. Ultimately, ES is able to find seven new best solutions that are shown in Table 5. Since the algorithms are tested on different processors, we use https://www.cpubenchmark.net/singleThread.html website in order to obtain the single thread benchmark results to be able to compare the reported CPU times in a fair manner. CPU Mark results for the CPUs used in each study are given in Table 6. We omit CP since it is an exact algorithm with time-limited runs and each instance is run for 30 min. Since the GVNS has the slowest CPU, we calculate the CPU time result multipliers by calculating the ratio between the GVNS and the other algorithms. We use the multipliers to adjust the average reported CPU times for each algorithm. Scatter plots of mean percent gaps and mean adjusted CPU times are given Fig. 8 for Solomon instances. Note that ILS100 and SAILS100 are not included in Fig. 8 since their average times (AT values) are not reported in Gunawan et al. (2017). For m = 1 in Fig. 8, it can be observed that the fastest algorithms are ILS, GRILS, ABC, ES, and SSA while ES weakly dominates them in terms of CPU time. The best algorithm is SAILSI3CH at the expense of CPU time. SAILSVNS performed similarly to ES at the expense of extensive CPU time. For m = 2, again, SAILSI3CH is the best performing algorithm in terms of Gap value at the expense of a very high CPU time. ES dominates both I3CH and SAILSVNS in terms of both Gap and CPU time. For m = 3, the I3CH is the best performing algorithm in terms of Gap value at the expense of a very high CPU time whereas ES dominates ABC, SSA, GVNS, SAILSI3CH, and SAILSVNS in terms of both Gap value and CPU time. For m = 4, the best algorithm is I3CH in terms of Gap value at the expense of a very high CPU time. ES weakly dominates SAILSI3CH and also dominates SSA, GVNS, and SAILSVNS in terms of both Gap value and CPU time. Scatter plots of mean percent gaps and mean adjusted CPU times are given in Fig. 9 for Cordeau instances. For m = 1 in Fig. 9, it can be observed that the best algorithm is SAILSI3CH in terms of Gap value with a reasonable CPU time. SAILSVNS is the second one at the expense of a very high CPU time. SAILSI3CH dominates SSA, ABC, and I3CH in terms of both Gap value and CPU time. ES and GVNS are very competitive with them. For m = 2, GVNS weakly dominates SAILSI3CH but they are the best performing algorithms in terms of Gap value. GVNS dominates ES, I3CH, ABC, SSA, and SAILSVNS. ES is very competitive to GVNS in terms of both Gap value and CPU time. For m = 3, GVNS and I3CH algorithms are the best ones in terms of Gap values but I3CH is computationally very expensive. ES weakly dominates SAILSI3CH but dominates SSA, ABC, and SAILSVNS. For m = 4, I3CH is the best performing algorithm in terms of Gap value at the expense of a very high CPU time. ES weakly dominates GVNS and SAILSI3CH but dominates SSA, ABC, and SAILSVNS. Note that the fastest algorithm is ILS. Scatter plots of mean percent gaps and mean adjusted CPU times are given in Fig. 10 for Vansteenwegen instances. For Solomon instances in Fig. 10, it can be observed that the best algorithm is I3CH in terms of Gap value with a very high CPU time. ES is the second in terms of Gap value and a reasonable CPU time. ILS is the fastest algorithm again. For Cordeau instances, I3CH is the best algorithm in terms of Gap value at the expense of a very high CPU time. ES is the second one with a very reasonable CPU time. ILS is again the fastest algorithm.

algorithm. The following components are analyzed below: ES without local search, Relocation (RL, both forward and backward) only, Only RL + N0, Only RL + N1, Only RL + N2, Only RL + N3, and Only RL + N4. We run all eight variants of the ES algorithm on all problem instances. The results are given in Table 7. It is clear from Table 7 that applying local search improves the solution quality since without the local search, the average gap was 1.05% and it is decreased up to 0.95% with the inclusion of the local search. Only RL + N3 component was the best one with a 0.92% gap but it cannot be applicable to instances with m = 1. In addition, Only RL + N1 was the second best one but again it cannot be applicable to instances with m = 1. Without these two components, the ES performed better than the remaining components by generating an average Gap of 0.95%. In order to see how it performs without m = 1, we provide additional average gap values in Table 7. It can be observed from Table 7 that the ES algorithm has generated 0.89% Gap with an average CPU time of 67.1 s, smaller CPU than all other components. 5. Conclusion A novel KT constructive heuristic and a self-adaptive evolution strategy for solving the TOPTW are presented in this paper. Three individuals in the initial population are constructed using KT, LS1 and VI heuristics. Then, the rest of the population is constructed by perturbations of these three heuristics in such a way that randomly chosen two neighborhood structures are applied to these three solutions. Offspring are generated by the ruin and recreate algorithm, where ruin size is adaptively updated by the evolution strategy. Four efficient and effective neighborhood structures are designed and used in a random local search scheme. In addition, the speed-up method and the proposed desirability measure are employed in the evolution strategy wherever necessary. For the selection of the population for the next generation, a goodness score is proposed to diversify the population. Extensive computational results are presented and compared to the best performing algorithms from the literature by using interval plots and scatter plots to see differences between algorithms. Computational results show that the evolution strategy is very competitive or even better than the competing algorithms. Ultimately, seven new best-known solutions are presented in the paper and they are given in Appendix B. Experimental results show that the proposed constructive heuristic can generate good and feasible solutions in a short time. The selfadaptive ES utilizing the ruin and recreate heuristic is a very fast and effective algorithm to solve TOPTW producing high-quality solutions while utilizing less CPU time when compared to the alternative methods. The proposed algorithm achieves very good results even without the local search procedure which shows the effectiveness of the algorithm. Swapping nodes between tours and moving a node to another tour neighborhoods are the better improvement moves among the four moves used in this study when there is more than one tour. In addition, the proposed self-adaptive ES is simple and can be easily adapted for other combinatorial problems using a problem specific ruin and recreate method. It has only two parameters that also makes adaption to new problems easy. Acknowledgement

4.4. Analysis of local search components

M. Fatih Tasgetiren acknowledges the HUST Project in Wuhan, China. He is partially supported by the National Natural Science Foundation of China with Grant No. 51435009.

We carried out experiments in order to analyze the impact of components of local search procedure on the performance of the ES Appendix A. Detailed results of the algorithms See Tables A1–A10.

11

BKS

320 360 400 420 340 340 370 370 380 870 930 960 980 910 930 930 950 198 286 293 303 247 293 299 308 277 284 297 298 797 930 1028 1093 953 1032 1077 1118 961 1000 1051 219 266 266 301

Prob.

c101 c102 c103 c104 c105 c106 c107 c108 c109 c201 c202 c203 c204 c205 c206 c207 c208 r101 r102 r103 r104 r105 r106 r107 r108 r109 r110 r111 r112 r201 r202 r203 r204 r205 r206 r207 r208 r209 r210 r211 rc101 rc102 rc103 rc104

320 360 400 420 340 340 370 370 380 860 930 960 970 900 930 930 950 198 286 293 303 247 293 297 308 277 284 297 298 795 923 1025 1090 953 1029 1077 1097 958 991 1049 219 266 266 301

Best

ES

320.0 360.0 400.0 420.0 340.0 340.0 370.0 370.0 380.0 860.0 930.0 960.0 970.0 900.0 928.0 930.0 948.0 198.0 286.0 293.0 303.0 247.0 293.0 297.0 308.0 277.0 284.0 297.0 298.0 795.0 922.0 1015.8 1081.8 953.0 1029.0 1077.0 1095.4 949.6 983.0 1048.4 219.0 266.0 266.0 301.0

Avg.

320 360 400 420 340 340 370 370 380 860 930 960 970 900 920 930 940 198 286 293 303 247 293 297 308 277 284 297 298 795 921 1009 1069 953 1029 1077 1094 939 974 1046 219 266 266 301

Worst

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.1 0.0 0.0 1.0 1.1 0.2 0.0 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.0 0.0 0.0 0.0 0.0 0.3 0.9 1.2 1.0 0.0 0.3 0.0 2.0 1.2 1.7 0.2 0.0 0.0 0.0 0.0

Gap

Table A1 Results for Solomon's problems for m = 1.

0.0 0.0 0.2 0.5 0.0 0.0 0.4 0.0 0.0 20.5 14.3 0.2 0.2 0.0 81.8 30.7 17.5 3.4 0.0 0.0 5.4 0.0 0.0 0.1 0.4 0.1 7.5 0.2 3.5 6.1 88.7 115.0 26.1 13.5 43.3 1.7 55.1 87.4 69.7 4.8 0.1 0.1 0.1 0.1

t (s) 320 360 390 400 340 340 360 370 380 870 930 960 960 910 920 920 940 198 286 293 303 247 293 297 306 277 284 297 291 792 872 962 1048 892 953 1012 1078 926 939 991 219 266 266 301

Best

CP

0.0 0.0 2.5 4.8 0.0 0.0 2.7 0.0 0.0 0.0 0.0 0.0 2.0 0.0 1.1 1.1 1.1 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.6 0.0 0.0 0.0 2.3 0.6 6.2 6.4 4.1 6.4 7.7 6.0 3.6 3.6 6.1 5.7 0.0 0.0 0.0 0.0

Gap 320 360 400 420 340 340 370 370 380 870 930 960 970 900 920 930 950 198 286 293 298 247 293 297 306 277 284 295 289 789 930 1020 1073 946 1021 1050 1092 948 982 1013 219 266 266 301

Best

I3CH

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.1 1.1 0.0 0.0 0.0 0.0 0.0 1.7 0.0 0.0 0.7 0.6 0.0 0.0 0.7 3.0 1.0 0.0 0.8 1.8 0.7 1.1 2.5 2.3 1.4 1.8 3.6 0.0 0.0 0.0 0.0

Gap 21.8 28.1 27.1 27.1 23.4 23.6 24.7 24.8 26.3 70.1 87.6 92.3 117.4 70.7 75.7 77.4 84.0 20.4 29.3 28.8 27.3 26.0 29.4 27.8 29.7 31.1 33.9 27.7 32.0 101.8 175.6 221.4 236.9 129.3 169.3 192.8 230.0 136.5 176.9 167.4 21.8 25.5 27.1 27.2

t (s) 320 360 400 410 340 340 360 370 380 850 930 960 970 900 930 930 950 197 281 288 301 236 283 291 308 277 281 292 290 785 890 1002 1077 919 982 1027 1086 933 975 1038 219 253 256 301

Best

GVNS

320.0 360.0 396.0 410.0 340.0 340.0 358.0 354.0 380.0 850.0 916.0 956.0 966.0 898.0 922.0 928.0 942.0 197.0 274.8 286.0 298.6 230.6 280.4 287.2 301.4 276.4 279.2 290.6 289.6 775.6 881.4 992.2 1074.0 905.8 966.6 1022.0 1084.0 926.0 961.4 1026.0 219.0 246.0 253.2 301.0

Avg. 0.0 0.0 0.0 2.4 0.0 0.0 2.7 0.0 0.0 2.3 0.0 0.0 1.0 1.1 0.0 0.0 0.0 0.5 1.7 1.7 0.7 4.5 3.4 2.7 0.0 0.0 1.1 1.7 2.7 1.5 4.3 2.5 1.5 3.6 4.8 4.6 2.9 2.9 2.5 1.2 0.0 4.9 3.8 0.0

Gap 0.2 67.4 382.4 1005.9 3.3 6.5 0.7 1.1 30.6 0.1 78.7 329.0 974.0 12.5 34.4 36.3 74.2 0.2 13.4 33.8 72.7 0.3 19.1 50.3 50.1 10.4 16.8 54.2 31.9 6.7 13.4 34.7 77.6 15.3 34.9 46.6 52.9 37.3 30.1 22.5 2.1 5.8 23.3 16.2

t (s) 320 360 400 420 340 340 370 370 380 870 930 960 970 910 930 930 950 198 286 293 303 247 293 297 306 277 284 297 298 794 914 997 1058 946 1020 1069 1079 945 973 1041 219 266 266 301

Best

SSA

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.6 0.0 0.0 0.0 0.0 0.4 1.7 3.0 3.2 0.7 1.2 0.7 3.5 1.7 2.7 1.0 0.0 0.0 0.0 0.0

Gap 20.4 20.7 24.2 21.9 20.6 20.3 20.5 20.5 20.5 28.3 33.5 59.6 42.3 46.9 29.0 29.1 31.2 19.7 21.0 20.3 23.2 20.3 22.1 21.5 40.6 20.4 20.8 27.9 22.3 51.1 46.4 44.2 39.7 37.8 40.9 52.5 35.8 61.6 53.6 40.5 19.8 20.2 20.7 27.5

t (s) 320 360 390 400 340 340 360 370 380 840 910 940 950 900 910 910 930 182 286 286 297 247 293 288 297 276 281 295 295 788 880 980 1073 931 996 1038 1069 926 958 1023 219 259 265 297

Best

ILS

0.0 0.0 2.5 4.8 0.0 0.0 2.7 0.0 0.0 3.4 2.2 2.1 3.1 1.1 2.2 2.2 2.1 8.1 0.0 2.4 2.0 0.0 0.0 3.7 3.6 0.4 1.1 0.7 1.0 1.1 5.4 4.7 1.8 2.3 3.5 3.6 4.4 3.6 4.2 2.7 0.0 2.6 0.4 1.3

Gap 0.4 0.3 0.5 0.3 0.3 0.3 0.3 0.3 0.3 1.1 2.8 1.7 1.6 1.2 1.6 2.1 1.6 0.1 0.2 0.2 0.2 0.1 0.2 0.2 0.2 0.2 0.3 0.2 0.2 1.2 1.4 1.6 1.7 1.4 1.5 2.0 1.6 2.4 1.9 1.6 0.2 0.2 0.3 0.3

t(s) 320 360 399 420 340 340 370 370 380 870 930 960 971 904 930 920 950 191 282 292 301 245 290 299 308 277 284 297 298 788 904 1016 1073 953 1021 1069 1094 944 966 1023 219 266 266 301

Best

ABC

320.0 360.0 398.0 420.0 340.0 340.0 370.0 370.0 380.0 870.0 930.0 958.0 966.0 900.0 928.0 918.0 950.0 190.0 281.6 292.0 299.6 242.6 289.0 299.0 308.0 277.0 282.2 294.0 297.0 787.0 895.8 1009.0 1070.4 953.0 1018.0 1060.8 1084.6 934.6 965.4 1019.0 219.0 266.0 266.0 301.0

Avg. 0.0 0.0 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 1.4 1.1 0.2 1.3 0.0 4.0 1.5 0.3 1.1 1.8 1.4 0.0 0.0 0.0 0.6 1.0 0.3 1.3 3.7 1.8 2.1 0.0 1.4 1.5 3.0 2.7 3.5 3.0 0.0 0.0 0.0 0.0

Gap 0.4 0.6 3.6 15.2 0.5 0.5 1.3 0.9 0.5 11.2 15.3 14.8 25.7 1.7 6.2 2.3 11.8 0.3 2.2 1.1 1.7 9.1 1.6 7.1 4.6 1.6 8.1 1.7 2.4 20.3 40.6 25.6 34.0 32.9 21.3 20.9 26.7 26.3 29.3 15.0 0.5 0.6 10.7 20.3

t (s)

12

320.0 360.0 397.0 411.0 340.0 340.0 370.0 370.0 380.0 867.0 919.0 943.0 961.0 907.0 920.0 921.0 943.0 198.0 286.0 293.0 302.6 247.0 293.0 299.0 308.0 277.0 281.4 296.4 293.7 786.2 894.6 989.6 1068.7 931.2 1005.1 1042.7 1084.0 914.4 955.1 1008.5 219.0 266.0 266.0 300.6

Avg. 0.0 0.0 0.8 2.1 0.0 0.0 0.0 0.0 0.0 0.3 1.2 1.8 1.9 0.3 1.1 1.0 0.7 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.9 0.2 1.4 1.4 3.8 3.7 2.2 2.3 2.6 3.2 3.0 4.8 4.5 4.0 0.0 0.0 0.0 0.1

Gap

1.2 1.7 2.4 2.9 1.1 1.3 1.4 1.5 1.8 3.1 5.5 8.3 11.0 4.2 5.1 5.7 6.1 0.7 1.4 1.7 2.0 0.9 1.5 1.7 2.0 1.2 1.3 1.7 1.5 4.6 7.1 11.4 17.5 7.5 9.6 13.5 19.8 9.2 9.8 11.3 0.7 1.0 1.2 1.4

t (s)

(continued on next page)

320 360 400 420 340 340 370 370 380 870 930 960 970 910 920 930 950 198 286 293 303 247 293 299 308 277 284 297 298 796 911 1005 1075 942 1014 1058 1088 924 965 1021 219 266 266 301

Best

GRILS

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

BKS

244 252 277 298 795 938 1003 1140 859 899 983 1057

Prob.

rc105 rc106 rc107 rc108 rc201 rc202 rc203 rc204 rc205 rc206 rc207 rc208

244 252 277 298 795 934 990 1122 859 895 983 1057

Best

ES

Table A1 (continued)

244.0 252.0 277.0 298.0 795.0 928.2 987.6 1117.8 858.4 890.8 983.0 1051.4

Avg.

244 252 277 298 795 905 982 1113 856 876 983 1050

Worst

0.0 0.0 0.0 0.0 0.0 1.0 1.5 1.9 0.1 0.9 0.0 0.5 0.3

Gap

3.2 0.5 0.0 0.0 4.8 27.8 62.3 162.2 1.1 24.9 2.3 5.8 17.7

t (s) 244 252 277 298 785 905 930 1066 812 874 952 1012

Best

CP

0.0 0.0 0.0 0.0 1.3 3.5 7.3 6.5 5.5 2.8 3.2 4.3 2.0

Gap 244 250 274 264 795 924 966 1093 847 863 957 1003

Best

I3CH

0.0 0.8 1.1 11.4 0.0 1.5 3.7 4.1 1.4 4.0 2.6 5.1 1.1

Gap 26.4 25.0 26.3 25.1 80.9 129.3 134.3 167.5 99.2 98.4 122.0 123.0 77.5

t (s) 231 252 274 298 788 900 969 1137 853 879 927 1014

Best

GVNS

227.0 252.0 261.2 288.8 784.0 890.6 954.4 1102.0 843.8 866.6 911.4 1000.0

Avg. 5.3 0.0 1.1 0.0 0.9 4.1 3.4 0.3 0.7 2.2 5.7 4.1 1.8

Gap 2.7 2.2 15.7 10.4 3.7 12.1 22.5 31.9 7.0 11.6 14.7 24.6 70.9

t (s) 244 252 277 298 795 930 967 1140 854 885 977 1041

Best

SSA

0.0 0.0 0.0 0.0 0.0 0.9 3.6 0.0 0.6 1.6 0.6 1.5 0.6

Gap 20.3 21.0 22.0 26.0 44.4 46.3 32.4 46.5 52.6 60.2 68.4 51.2 33.1

t (s) 221 239 274 288 780 882 960 1117 840 860 926 1037

Best

ILS

9.4 5.2 1.1 3.4 1.9 6.0 4.3 2.0 2.2 4.3 5.8 1.9 2.5

Gap 0.2 0.2 0.2 0.2 1.0 1.3 2.7 2.3 1.0 1.1 1.3 2.3 0.9

t(s) 244 252 277 298 789 931 967 1116 854 895 977 1042

Best

ABC

244.0 251.4 276.2 298.0 784.0 926.8 962.4 1109.2 852.3 890.2 977.0 1032.8

Avg. 0.0 0.2 0.3 0.0 1.4 1.2 4.0 2.7 0.8 1.0 0.6 2.3 1.0

Gap 1.3 8.1 10.4 11.3 24.9 27.3 24.8 23.4 27.2 24.9 15.8 26.3 12.6

t (s) 244 252 277 298 784 917 966 1121 855 877 960 1024

Best

GRILS

244.0 249.3 276.4 295.5 781.8 909.8 960.5 1103.0 847.2 866.2 930.8 1004.0

Avg. 0.0 1.1 0.2 0.8 1.7 3.0 4.2 3.2 1.4 3.6 5.3 5.0 1.4

Gap

0.9 0.9 1.1 1.1 3.5 5.8 8.8 14.6 4.6 5.5 7.0 7.9 4.8

t (s)

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

13

BKS

590 660 720 760 640 620 670 680 720 1460 1470 1480 1490 1470 1480 1490 1490 349 508 522 552 453 529 538 560 506 525 544 544 1256 1348 1418 1458 1386 1450 1458 1458 1414 1427 1458 427 505 524 575

Prob.

c101 c102 c103 c104 c105 c106 c107 c108 c109 c201 c202 c203 c204 c205 c206 c207 c208 r101 r102 r103 r104 r105 r106 r107 r108 r109 r110 r111 r112 r201 r202 r203 r204 r205 r206 r207 r208 r209 r210 r211 rc101 rc102 rc103 rc104

590 660 720 760 640 620 670 680 720 1450 1470 1470 1480 1470 1480 1480 1480 349 508 520 549 447 529 538 560 506 525 541 544 1254 1347 1416 1458 1383 1440 1458 1458 1417 1420 1458 427 505 523 574

Best

ES

590.0 660.0 718.0 760.0 640.0 620.0 670.0 680.0 720.0 1450.0 1460.0 1466.0 1478.0 1470.0 1480.0 1480.0 1480.0 349.0 508.0 520.0 547.2 447.0 529.0 538.0 558.8 506.0 525.0 541.0 542.2 1247.2 1341.6 1413.8 1458.0 1380.2 1438.2 1455.4 1458.0 1404.0 1415.4 1457.0 427.0 503.4 523.0 573.0

Avg.

590 660 710 760 640 620 670 680 720 1450 1450 1460 1470 1470 1480 1480 1480 349 508 520 546 447 529 538 558 506 525 541 535 1241 1336 1408 1458 1379 1436 1453 1458 1394 1411 1455 427 497 523 569

Worst

0.0 0.0 0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.7 0.9 0.8 0.0 0.0 0.7 0.7 0.0 0.0 0.4 0.9 1.3 0.0 0.0 0.2 0.0 0.0 0.6 0.3 0.7 0.5 0.3 0.0 0.4 0.8 0.2 0.0 0.7 0.8 0.1 0.0 0.3 0.2 0.3

Gap

Table A2 Results for Solomon's problems for m = 2.

0.2 135.2 37.5 0.3 0.0 0.0 0.0 0.1 0.2 2.6 67.6 12.5 47.2 93.1 12.5 29.8 2.5 0.1 0.3 2.1 3.3 2.4 0.2 0.2 39.0 2.1 2.4 4.3 14.8 107.2 103.3 55.7 43.3 138.7 128.9 89.6 0.4 157.3 107.4 94.6 0.1 63.7 1.0 19.5

t (s) 590 660 720 760 640 620 670 680 720 1460 1470 1470 1480 1470 1480 1480 1480 349 470 515 537 453 529 537 558 505 500 520 526 1246 1348 1413 1452 1371 1433 1443 1458 1391 1407 1451 427 504 505 575

Best

CP

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.7 0.0 0.0 0.7 0.7 0.0 7.5 1.3 2.7 0.0 0.0 0.2 0.4 0.2 4.8 4.4 3.3 0.8 0.0 0.4 0.4 1.1 1.2 1.0 0.0 1.6 1.4 0.5 0.0 0.2 3.6 0.0

Gap 590 660 720 760 640 620 670 680 720 1450 1470 1470 1480 1450 1480 1470 1470 349 508 519 549 447 529 533 550 506 525 542 534 1254 1344 1416 1458 1380 1427 1458 1458 1404 1415 1450 427 505 519 556

Best

I3CH

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.0 0.7 0.7 1.4 0.0 1.3 1.3 0.0 0.0 0.6 0.5 1.3 0.0 0.9 1.8 0.0 0.0 0.4 1.8 0.2 0.3 0.1 0.0 0.4 1.6 0.0 0.0 0.7 0.8 0.5 0.0 0.0 1.0 3.3

Gap 53.4 78.4 116.1 94.4 70.6 149.2 65.4 85.9 69.4 321.2 405.9 458.2 498.5 322.2 355.9 423.5 424.3 42.1 62.4 68.3 75.3 56.2 63.5 63.2 65.7 60.5 68.9 67.0 63.0 333.6 588.5 815.9 33.5 606.1 739.9 453.8 4.3 600.3 728.5 890.9 52.3 59.8 60.3 59.9

t (s) 590 650 710 750 640 620 670 680 720 1450 1470 1470 1480 1470 1480 1490 1490 349 499 508 541 445 529 527 551 506 523 533 542 1235 1327 1407 1458 1362 1435 1454 1458 1402 1404 1455 427 504 514 562

Best

GVNS

590.0 648.0 704.0 746.0 640.0 620.0 670.0 680.0 716.0 1450.0 1464.0 1464.0 1472.0 1466.0 1472.0 1484.0 1480.0 349.0 492.8 506.6 539.2 442.6 524.4 522.2 545.6 501.6 523.0 530.6 536.6 1225.8 1306.2 1392.6 1452.6 1351.6 1429.8 1449.8 1458.0 1389.6 1391.0 1452.6 420.6 485.4 502.0 554.8

Avg. 0.0 1.5 1.4 1.3 0.0 0.0 0.0 0.0 0.0 0.7 0.0 0.7 0.7 0.0 0.0 0.0 0.0 0.0 1.8 2.7 2.0 1.8 0.0 2.0 1.6 0.0 0.4 2.0 0.4 1.7 1.6 0.8 0.0 1.7 1.0 0.3 0.0 0.8 1.6 0.2 0.0 0.2 1.9 2.3

Gap 9.3 47.5 271.6 616.4 35.0 44.4 34.8 65.7 131.1 5.8 21.1 46.1 135.5 10.8 14.1 18.1 18.8 4.0 17.4 37.4 93.7 5.3 31.2 56.5 179.5 19.7 65.5 115.1 98.8 15.0 15.4 17.7 8.9 20.1 17.1 12.4 8.3 16.3 16.0 14.8 2.8 8.4 26.8 43.6

t (s) 590 660 720 760 640 620 670 680 720 1450 1460 1450 1450 1470 1460 1480 1460 349 508 519 548 453 529 532 558 506 525 544 542 1242 1334 1414 1447 1363 1430 1452 1457 1404 1414 1451 427 505 523 575

Best

SSA

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.7 2.0 2.7 0.0 1.4 0.7 2.0 0.0 0.0 0.6 0.7 0.0 0.0 1.1 0.4 0.0 0.0 0.0 0.4 1.1 1.0 0.3 0.8 1.7 1.4 0.4 0.1 0.7 0.9 0.5 0.0 0.0 0.2 0.0

Gap 26.2 26.5 26.8 28.0 26.1 25.6 26.3 26.2 26.1 64.3 50.2 39.4 38.9 68.0 42.4 87.7 38.4 25.0 43.9 32.3 29.1 25.4 27.2 37.0 67.5 45.5 27.5 54.0 25.1 73.7 50.4 196.2 70.9 51.5 113.7 158.8 56.0 89.5 77.8 66.9 25.3 42.4 26.5 63.1

t (s) 590 650 700 750 640 620 670 670 710 1400 1430 1430 1460 1450 1440 1450 1460 330 508 513 539 430 529 529 549 498 515 535 515 1231 1270 1377 1440 1338 1401 1428 1458 1345 1365 1422 427 494 519 565

Best

ILS

0.0 1.5 2.8 1.3 0.0 0.0 0.0 1.5 1.4 4.1 2.7 3.4 2.0 1.4 2.7 2.7 2.0 5.4 0.0 1.7 2.4 5.1 0.0 1.7 2.0 1.6 1.9 1.7 5.3 2.0 5.8 2.9 1.2 3.5 3.4 2.1 0.0 4.9 4.3 2.5 0.0 2.2 1.0 1.7

Gap 1.4 0.9 1.2 1.5 0.8 0.8 1.4 0.8 0.9 2.7 5.1 3.8 4.2 3.1 2.8 3.2 2.8 0.4 0.9 0.9 1.5 0.8 0.9 1.0 1.4 0.5 1.0 0.6 0.5 2.1 2.3 1.9 3.4 2.8 2.8 1.7 1.6 2.6 1.9 1.9 0.6 0.8 1.1 0.7

T(s) 590 660 720 760 640 620 670 680 720 1460 1460 1457 1460 1470 1470 1480 1479 345 508 519 549 450 529 520 557 504 525 544 544 1239 1344 1414 1457 1375 1430 1455 1458 1400 1419 1457 427 505 524 575

Best

ABC

590.0 660.0 716.0 758.0 640.0 620.0 670.0 680.0 720.0 1450.0 1458.0 1450.0 1450.0 1470.0 1470.0 1477.0 1472.0 342.8 508.0 517.2 549.0 448.0 529.0 518.8 556.0 502.8 525.0 544.0 544.0 1231.2 1341.6 1410.0 1453.4 1365.8 1426.8 1448.4 1457.8 1390.2 1414.2 1451.2 425.8 505.0 523.0 572.8

Avg. 0.0 0.0 0.6 0.3 0.0 0.0 0.0 0.0 0.0 0.7 0.8 2.0 2.7 0.0 0.7 0.9 1.2 1.8 0.0 0.9 0.5 1.1 0.0 3.6 0.7 0.6 0.0 0.0 0.0 2.0 0.5 0.6 0.3 1.5 1.6 0.7 0.0 1.7 0.9 0.5 0.3 0.0 0.2 0.4

Gap 12.3 12.4 19.6 13.5 2.1 2.3 2.1 3.1 8.9 22.4 32.2 15.9 17.5 24.7 3.8 28.0 9.2 2.9 26.1 19.1 17.3 20.7 11.1 21.9 13.7 8.9 12.4 11.8 11.5 23.0 40.2 43.7 35.6 47.1 48.4 35.7 12.6 35.0 38.2 29.3 5.4 5.1 8.7 26.2

t (s)

14

589.0 649.0 693.0 729.0 640.0 620.0 670.0 677.0 705.0 1421.0 1415.0 1420.0 1437.0 1435.0 1437.0 1441.0 1446.0 349.0 504.8 516.0 538.6 453.0 522.5 529.9 550.9 498.0 501.9 534.8 523.1 1212.3 1289.5 1372.0 1433.1 1328.6 1388.9 1422.9 1455.4 1344.3 1365.2 1402.3 427.0 501.8 515.1 555.0

Avg. 0.2 1.7 3.8 4.1 0.0 0.0 0.0 0.4 2.1 2.7 3.7 4.1 3.6 2.4 2.9 3.3 3.0 0.0 0.6 1.1 2.4 0.0 1.2 1.5 1.6 1.6 4.4 1.7 3.8 3.5 4.3 3.2 1.7 4.1 4.2 2.4 0.2 4.9 4.3 3.8 0.0 0.6 1.7 3.5

Gap

3.2 4.7 6.0 7.4 3.6 3.9 4.2 4.5 5.2 8.7 11.8 15.5 20.6 10.3 11.8 12.2 13.2 2.2 3.6 4.4 5.1 2.9 3.8 4.6 5.2 3.6 3.9 4.4 4.1 9.6 11.0 14.7 18.9 13.3 14.3 17.7 20.2 14.4 14.1 16.5 2.5 3.2 3.5 3.9

t (s)

(continued on next page)

590 650 710 730 640 620 670 680 710 1430 1420 1430 1440 1440 1440 1450 1450 349 508 517 541 453 525 535 558 498 508 539 536 1236 1298 1383 1440 1344 1398 1431 1458 1360 1371 1413 427 504 523 564

Best

GRILS

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

BKS

480 483 534 556 1385 1512 1632 1716 1458 1552 1599 1692

Prob.

rc105 rc106 rc107 rc108 rc201 rc202 rc203 rc204 rc205 rc206 rc207 rc208

480 483 529 553 1375 1511 1629 1706 1458 1538 1587 1691

Best

ES

Table A2 (continued)

480.0 483.0 529.0 553.0 1369.6 1503.6 1605.8 1701.6 1427.4 1521.8 1569.0 1683.4

Avg.

480 483 529 553 1355 1492 1589 1697 1388 1492 1554 1667

Worst

0.0 0.0 0.9 0.5 1.1 0.6 1.6 0.8 2.1 1.9 1.9 0.5 0.5

Gap 78.8 64.5 7.3 5.9 84.1 65.3 101.1 115.6 105.8 112.5 111.6 190.6 47.6

t (s) 465 480 524 546 1310 1494 1534 1660 1458 1469 1547 1636

Best

CP

3.1 0.6 1.9 1.8 5.4 1.2 6.0 3.3 0.0 5.3 3.3 3.3 1.3

Gap 480 481 529 547 1384 1500 1627 1704 1452 1525 1582 1669

Best

I3CH

0.0 0.4 0.9 1.6 0.1 0.8 0.3 0.7 0.4 1.7 1.1 1.4 0.6

Gap 58.6 56.0 62.9 61.4 267.4 386.7 482.8 760.8 311.0 335.9 408.6 564.4 259.5

t (s) 478 483 529 553 1369 1476 1570 1713 1407 1531 1563 1687

Best

GVNS

456.8 480.4 524.0 544.6 1359.6 1465.0 1561.6 1699.2 1396.0 1507.4 1546.6 1669.6

Avg. 0.4 0.0 0.9 0.5 1.2 2.4 3.8 0.2 3.5 1.4 2.3 0.3 0.9

Gap 7.1 11.0 26.0 36.8 7.3 16.8 11.8 8.8 8.5 18.1 18.3 12.5 47.8

t (s) 480 483 529 554 1377 1499 1576 1674 1458 1528 1574 1675

Best

SSA

0.0 0.0 0.9 0.4 0.6 0.9 3.4 2.4 0.0 1.5 1.6 1.0 0.6

Gap 56.3 47.3 26.8 36.1 73.3 51.8 94.9 115.6 50.5 50.8 73.5 130.4 54.9

t (s) 459 458 515 546 1305 1461 1573 1656 1381 1495 1531 1606

Best

ILS

4.4 5.2 3.6 1.8 5.8 3.4 3.6 3.5 5.3 3.7 4.3 5.1 2.6

Gap 0.8 0.6 0.5 0.6 1.9 2.1 2.0 2.1 3.2 1.9 2.7 1.7 1.7

T(s) 476 483 534 556 1377 1475 1594 1688 1449 1506 1567 1653

Best

ABC

475.2 483.0 531.4 554.0 1373.2 1472.6 1593.6 1682.0 1446.0 1503.8 1561.8 1647.4

Avg. 1.0 0.0 0.5 0.4 0.9 2.6 2.4 2.0 0.8 3.1 2.3 2.6 0.9

Gap 28.2 12.6 19.8 17.3 18.5 36.8 42.0 49.6 26.2 28.1 37.7 46.1 21.5

t (s) 478 481 520 537 1345 1480 1560 1656 1401 1505 1506 1620

Best

GRILS

476.2 472.3 512.0 525.2 1327.4 1441.0 1535.6 1642.0 1383.8 1460.1 1492.4 1584.6

Avg. 0.8 2.2 4.1 5.5 4.2 4.7 5.9 4.3 5.1 5.9 6.7 6.3 2.8

Gap

2.9 2.9 3.2 3.3 8.7 11.0 14.6 18.5 10.2 11.4 12.6 15.0 8.7

t (s)

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

15

BKS

810 920 990 1030 870 870 910 920 970 1810 1810 1810 1810 1810 1810 1810 1810 484 694 747 778 620 729 760 797 710 737 774 776 1442 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 621 714 764 835

Prob.

c101 c102 c103 c104 c105 c106 c107 c108 c109 c201 c202 c203 c204 c205 c206 c207 c208 r101 r102 r103 r104 r105 r106 r107 r108 r109 r110 r111 r112 r201 r202 r203 r204 r205 r206 r207 r208 r209 r210 r211 rc101 rc102 rc103 rc104

810 920 980 1030 870 870 910 920 970 1810 1810 1810 1810 1810 1810 1810 1810 481 691 740 778 620 729 759 797 710 736 774 776 1440 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 621 710 764 834

Best

ES

810.0 920.0 980.0 1022.0 870.0 870.0 910.0 916.0 964.0 1806.0 1808.0 1808.0 1810.0 1810.0 1810.0 1810.0 1810.0 481.0 691.0 740.0 778.0 619.2 728.0 759.0 797.0 706.4 736.0 773.2 775.6 1435.8 1457.4 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 616.8 710.0 757.0 833.2

Avg.

810 920 980 1020 870 870 910 910 960 1800 1800 1800 1810 1810 1810 1810 1810 481 691 740 778 616 724 759 797 701 736 773 774 1432 1455 1458 1458 1458 1458 1458 1458 1458 1458 1458 614 710 747 832

Worst

0.0 0.0 1.0 0.8 0.0 0.0 0.0 0.4 0.6 0.2 0.1 0.1 0.0 0.0 0.0 0.0 0.0 0.6 0.4 0.9 0.0 0.1 0.1 0.1 0.0 0.5 0.1 0.1 0.1 0.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.6 0.9 0.2

Gap

Table A3 Results for Solomon's problems for m = 3.

1.0 16.1 32.2 6.0 24.9 26.6 3.1 46.0 4.2 22.2 18.8 37.8 51.7 4.8 2.5 1.9 1.3 0.1 4.9 7.6 90.3 1.8 9.3 38.7 2.8 5.5 9.9 45.6 72.8 58.4 21.6 0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 5.9 10.0 13.7 29.8

t (s) 810 920 970 1020 860 870 900 920 960 1810 1810 1810 1810 1810 1810 1810 1810 484 662 716 754 615 716 739 771 691 720 756 749 1440 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 621 703 702 835

Best

CP

0.0 0.0 2.0 1.0 1.1 0.0 1.1 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 4.6 4.1 3.1 0.8 1.8 2.8 3.3 2.7 2.3 2.3 3.5 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.5 8.1 0.0

Gap 810 920 990 1030 870 870 910 920 960 1810 1810 1810 1810 1810 1810 1810 1810 481 691 740 777 619 729 759 797 710 736 773 776 1439 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 621 714 764 834

Best

I3CH

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.6 0.4 0.9 0.1 0.2 0.0 0.1 0.0 0.0 0.1 0.1 0.0 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1

Gap 303.1 237.5 156.5 155.9 110.2 227.4 151.8 212.6 157.2 3.8 7.2 7.9 11.2 12.6 18.0 16.9 20.6 67.7 111.4 125.6 128.3 165.8 122.7 120.5 135.7 103.6 109.9 118.1 110.5 981.2 15.3 0.2 0.2 0.3 0.3 0.3 0.2 0.3 0.2 0.4 87.6 105.4 105.7 124.4

t (s) 810 920 970 1020 870 870 910 920 960 1810 1810 1810 1810 1810 1810 1810 1810 481 686 730 770 615 723 759 781 700 734 761 757 1423 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 614 705 764 834

Best

GVNS

808 916 964 1012 864 866 906 914 958 1800 1806 1804 1802 1810 1810 1810 1810 477.4 674.2 724.4 762 612.4 713.4 755 770.4 699.6 716.2 749.4 751 1416.4 1450.2 1458 1458 1458 1458 1458 1458 1458 1458 1458 602.4 692 741.2 826

Avg 0.0 0.0 2.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.6 1.2 2.3 1.0 0.8 0.8 0.1 2.0 1.4 0.4 1.7 2.4 1.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.1 1.3 0.0 0.1

Gap 18.0 94.1 332.3 604.0 36.5 50.2 63.4 105.9 180.7 3.7 8.8 10.9 10.9 4.9 6.5 7.6 8.3 7.7 21.9 68.4 103.7 11.0 47.0 110.1 187.9 24.1 69.8 62.6 173.0 6.4 21.0 9.0 11.9 9.0 9.0 0.2 0.1 0.2 9.7 0.2 5.7 14.0 47.0 85.6

t (s) 810 920 980 1010 870 870 910 920 970 1800 1790 1770 1750 1810 1780 1800 1800 484 694 736 777 619 729 759 789 702 734 771 776 1429 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 621 710 764 814

Best

SSA

0.0 0.0 1.0 1.9 0.0 0.0 0.0 0.0 0.0 0.6 1.1 2.2 3.3 0.0 1.7 0.6 0.6 0.0 0.0 1.5 0.1 0.2 0.0 0.1 1.0 1.1 0.4 0.4 0.0 0.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.6 0.0 2.5

Gap 31.7 32.7 37.3 33.1 32.9 39.6 33.4 33.1 43.5 48.1 68.3 111.9 54.5 42.5 40.8 59.5 52.2 48.6 43.5 46.1 71.2 36.5 38.6 90.8 81.8 40.2 36.8 46.9 91.8 50.3 59.0 39.9 38.4 39.3 38.9 38.5 38.4 40.1 40.0 38.6 34.2 41.3 44.6 33.0

t (s) 790 890 960 1010 840 840 900 900 950 1750 1750 1760 1780 1770 1770 1810 1810 481 685 720 765 609 719 747 790 699 711 764 758 1408 1443 1458 1458 1458 1458 1458 1458 1458 1458 1458 604 698 747 822

Best

ILS

2.5 3.3 3.0 1.9 3.4 3.4 1.1 2.2 2.1 3.3 3.3 2.8 1.7 2.2 2.2 0.0 0.0 0.6 1.3 3.6 1.7 1.8 1.4 1.7 0.9 1.5 3.5 1.3 2.3 2.4 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.7 2.2 2.2 1.6

Gap 1.1 2.1 2.2 1.3 1.0 1.1 1.5 1.2 2.0 2.2 2.0 2.0 1.5 2.5 1.5 3.4 2.4 0.8 1.0 2.0 1.5 2.3 2.1 1.1 3.1 1.8 1.4 1.8 1.1 2.4 2.7 1.6 1.0 1.1 1.1 1.0 0.8 1.1 1.2 1.0 1.4 1.3 1.1 1.3

t(s) 810 920 980 1014 870 870 910 920 966 1810 1793 1774 1762 1810 1810 1810 1810 480 691 747 777 620 726 760 792 710 737 773 772 1441 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 621 713 754 831

Best

ABC

810.0 920.0 976.0 1008.0 868.0 870.0 910.0 920.0 962.0 1808.0 1792.0 1766.0 1758.0 1804.0 1810.0 1810.0 1810.0 477.2 690.6 746.0 777.0 619.3 725.8 760.0 785.8 704.0 734.0 771.0 770.8 1434.8 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 617.4 710.3 748.6 828.0

Avg 0.0 0.0 1.4 2.1 0.2 0.0 0.0 0.0 0.8 0.1 1.0 2.4 2.9 0.3 0.0 0.0 0.0 1.4 0.5 0.1 0.1 0.1 0.4 0.0 1.4 0.8 0.4 0.4 0.7 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.6 0.5 2.0 0.8

Gap 14.7 5.8 16.4 31.5 14.3 16.5 15.4 14.0 18.2 35.4 39.8 45.3 35.8 43.5 1.3 5.1 2.0 11.4 19.4 39.2 25.7 24.9 24.5 24.2 27.0 25.9 29.7 20.8 28.2 39.7 5.8 0.8 0.4 1.3 0.6 0.4 0.2 0.9 0.6 0.4 23.5 26.3 16.5 25.9

t (s) 810 900 960 990 860 860 890 900 940 1770 1730 1740 1760 1750 1770 1750 1770 484 686 730 761 619 715 745 778 702 704 755 744 1420 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 616 710 742 827

Best

16

806 888 942 977 849 853 883 892 925 1728 1724 1726 1753 1735 1740 1739 1751 482.4 676.1 716.8 752.4 614.4 710.1 735 771.2 697.4 686.8 746.1 731.2 1409.6 1449.1 1458 1458 1458 1458 1458 1458 1458 1458 1458 613 703 733.1 807.4

Avg 0.5 3.5 4.8 5.1 2.4 2.0 3.0 3.0 4.6 4.5 4.8 4.6 3.1 4.1 3.9 3.9 3.3 0.3 2.6 4.0 3.3 0.9 2.6 3.3 3.2 1.8 6.8 3.6 5.8 2.2 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.3 1.5 4.0 3.3

Gap

5.2 7.0 9.4 12.0 5.9 6.4 6.8 7.2 8.4 11.5 13.7 16.0 20.0 13.0 14.3 14.8 16.1 3.7 5.7 7.0 8.2 4.7 6.2 7.3 8.2 5.8 6.2 6.9 6.4 11.2 11.6 13.2 14.7 13.2 13.6 15.0 16.1 14.3 13.6 15.7 4.2 4.9 5.7 6.2

t (s)

(continued on next page)

GRILS

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

BKS

682 706 773 795 1698 1724 1724 1724 1719 1724 1724 1724

Prob.

rc105 rc106 rc107 rc108 rc201 rc202 rc203 rc204 rc205 rc206 rc207 rc208

682 706 768 795 1698 1724 1724 1724 1709 1724 1724 1724

Best

ES

Table A3 (continued)

682.0 705.8 768.0 792.8 1685.8 1720.4 1724.0 1724.0 1705.4 1724.0 1724.0 1724.0

Avg.

682 705 768 789 1668 1714 1724 1724 1691 1724 1724 1724

Worst

0.0 0.0 0.6 0.3 0.7 0.2 0.0 0.0 0.8 0.0 0.0 0.0 0.2

Gap 2.4 17.8 17.7 77.5 94.7 89.9 0.7 0.0 64.7 0.6 0.5 0.0 19.6

t (s) 666 693 759 769 1693 1724 1724 1724 1709 1724 1724 1724

Best

CP

2.3 1.8 1.8 3.3 0.3 0.0 0.0 0.0 0.6 0.0 0.0 0.0 1.0

Gap 682 706 762 789 1693 1724 1724 1724 1719 1724 1724 1724

Best

I3CH

0.0 0.0 1.4 0.8 0.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1

Gap 90.4 88.1 98.7 107.4 575.2 1.1 0.3 0.3 734.4 0.5 0.3 0.4 113.4

t (s) 680 695 762 782 1682 1709 1724 1724 1706 1724 1724 1724

Best

GVNS

670.2 692.6 756.4 773 1676 1702.8 1724 1724 1702.2 1724 1724 1724

Avg 0.3 1.6 1.4 1.6 0.9 0.9 0.0 0.0 0.8 0.0 0.0 0.0 0.5

Gap 20.8 17.4 30.7 48.2 11.6 10.5 7.9 0.1 11.6 8.7 8.7 0.2 50.7

t (s) 682 706 773 778 1681 1714 1724 1724 1709 1724 1724 1724

Best

SSA

0.0 0.0 0.0 2.1 1.0 0.6 0.0 0.0 0.6 0.0 0.0 0.0 0.5

Gap 32.0 61.5 43.8 52.0 98.3 48.3 39.1 38.3 122.3 46.0 40.4 39.1 49.0

t (s) 654 678 745 757 1625 1686 1724 1724 1659 1708 1713 1724

Best

ILS

4.1 4.0 3.6 4.8 4.3 2.2 0.0 0.0 3.5 0.9 0.6 0.0 1.8

Gap 0.8 1.0 0.9 1.1 1.9 1.7 2.9 1.0 2.4 1.3 1.5 1.1 1.6

t(s) 682 696 772 782 1689 1724 1724 1724 1709 1724 1724 1724

Best

ABC

682.0 696.4 769.8 781.2 1681.0 1717.4 1724.0 1724.0 1704.8 1724.0 1724.0 1724.0

Avg 0.0 1.4 0.4 1.7 1.0 0.4 0.0 0.0 0.8 0.0 0.0 0.0 0.5

Gap 28.7 24.4 34.7 26.2 32.1 12.7 2.1 1.1 22.7 4.3 1.9 1.1 17.7

t (s) 681 687 759 763 1657 1708 1724 1724 1687 1724 1724 1724

Best

GRILS

670 676.7 739.7 754.1 1638 1691.4 1724 1724 1669.2 1713.1 1714.8 1724

Avg 1.8 4.2 4.3 5.1 3.5 1.9 0.0 0.0 2.9 0.6 0.5 0.0 2.5

Gap

4.9 4.6 5.1 5.3 11.2 12.3 14.2 15.3 12.2 13.7 14.2 15.8 10.1

t (s)

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

17

BKS

1020 1150 1210 1260 1070 1080 1120 1140 1190 1810 1810 1810 1810 1810 1810 1810 1810 611 843 928 975 778 906 950 995 885 915 953 974 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 811 909 975 1065 875 909 987 1025 1724 1724 1724 1724

Prob.

c101 c102 c103 c104 c105 c106 c107 c108 c109 c201 c202 c203 c204 c205 c206 c207 c208 r101 r102 r103 r104 r105 r106 r107 r108 r109 r110 r111 r112 r201 r202 r203 r204 r205 r206 r207 r208 r209 r210 r211 rc101 rc102 rc103 rc104 rc105 rc106 rc107 rc108 rc201 rc202 rc203 rc204

1020 1140 1210 1260 1060 1080 1110 1130 1190 1810 1810 1810 1810 1810 1810 1810 1810 608 837 928 972 778 906 950 991 885 915 945 971 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 808 902 972 1065 875 909 980 1023 1724 1724 1724 1724

Best

ES

1020.0 1136.0 1196.0 1254.0 1058.0 1068.0 1110.0 1126.0 1184.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 608.0 837.0 928.0 969.6 776.6 903.8 947.2 991.0 885.0 907.6 944.4 967.4 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 808.0 902.0 971.0 1064.6 875.0 908.8 975.8 1021.6 1724.0 1724.0 1724.0 1724.0

Avg.

1020 1130 1180 1250 1050 1060 1110 1120 1180 1810 1810 1810 1810 1810 1810 1810 1810 608 837 928 969 775 895 945 991 885 900 944 964 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 808 902 970 1063 875 908 972 1020 1724 1724 1724 1724

Worst

0.0 1.2 1.2 0.5 1.1 1.1 0.9 1.2 0.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.7 0.0 0.6 0.2 0.2 0.3 0.4 0.0 0.8 0.9 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.8 0.4 0.0 0.0 0.0 1.1 0.3 0.0 0.0 0.0 0.0

Gap

Table A4 Results for Solomon's problems for m = 4.

53.0 51.6 49.2 53.4 18.8 63.2 16.8 98.8 55.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 14.0 47.9 5.5 51.5 147.2 24.7 57.5 79.3 131.5 50.8 12.2 69.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6 38.1 103.6 107.5 64.5 37.6 75.9 60.8 0.1 0.0 0.0 0.0

t (s) 1020 1140 1180 1240 1060 1070 1110 1120 1180 1810 1810 1810 1810 1810 1810 1810 1810 606 843 914 951 767 868 899 964 837 898 908 956 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 811 906 896 1017 867 826 980 1002 1724 1724 1724 1724

Best

CP

0.0 0.9 2.5 1.6 0.9 0.9 0.9 1.8 0.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.8 0.0 1.5 2.5 1.4 4.2 5.4 3.1 5.4 1.9 4.7 1.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 8.1 4.5 0.9 9.1 0.7 2.2 0.0 0.0 0.0 0.0

Gap 1020 1150 1210 1260 1060 1080 1120 1130 1190 1810 1810 1810 1810 1810 1810 1810 1810 608 837 928 969 778 906 950 994 885 915 952 967 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 808 899 974 1064 875 909 980 1020 1724 1724 1724 1724

Best

I3CH

0.0 0.0 0.0 0.0 0.9 0.0 0.0 0.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 0.7 0.0 0.6 0.0 0.0 0.0 0.1 0.0 0.0 0.1 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 1.1 0.1 0.1 0.0 0.0 0.7 0.5 0.0 0.0 0.0 0.0

Gap 176.4 274.1 301.1 281.8 333.2 179.4 209.3 373.1 227.6 0.1 0.1 0.2 0.1 0.1 0.1 0.1 0.1 93.5 162.6 206.9 209.1 142.4 185.9 221.4 220 145.8 168.6 203.5 251.8 0.2 0.2 0.2 0.2 0.2 0.1 0.2 0.2 0.2 0.2 0.2 119.7 136.7 170.2 194.1 127 143 155.4 173.2 0.2 0.2 0.2 0.2

t (s) 1020 1140 1180 1230 1060 1060 1120 1130 1180 1810 1810 1810 1810 1810 1810 1810 1810 608 822 914 939 776 884 946 968 879 915 952 968 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 808 897 953 1048 875 902 966 1023 1724 1724 1724 1724

Best

GVNS

1014 1132 1178 1226 1060 1052 1112 1116 1170 1810 1810 1810 1810 1810 1810 1810 1810 607 814.6 902 939 770.6 878.2 932.8 958.6 869.4 893.2 937.4 950.4 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 797.6 887 948.4 1036 866 896.2 963 1015 1723 1724 1724 1724

Avg 0.0 0.9 2.5 2.4 0.9 1.9 0.0 0.9 0.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 2.5 1.5 3.7 0.3 2.4 0.4 2.7 0.7 0.0 0.1 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 1.3 2.3 1.6 0.0 0.8 2.1 0.2 0.0 0.0 0.0 0.0

Gap 22.1 58.2 221.2 513.0 33.3 51.5 62.0 94.5 143.2 0.1 1.9 1.5 0.5 0.1 0.1 0.1 0.1 11.9 23.0 68.9 103.0 15.5 54.7 112.4 198.5 55.0 92.9 96.5 184.6 1.4 0.1 0.2 0.1 0.1 0.2 0.2 0.1 0.2 0.2 0.2 10.1 21.3 46.7 76.6 20.3 20.7 46.0 53.6 1.9 0.8 0.1 0.1

t (s) 1020 1150 1190 1230 1060 1080 1120 1130 1190 1810 1810 1810 1810 1810 1810 1810 1810 611 843 926 964 771 905 942 977 885 893 948 958 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 808 902 970 1059 875 901 980 1023 1724 1724 1724 1724

Best

SSA

0.0 0.0 1.7 2.4 0.9 0.0 0.0 0.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 1.1 0.9 0.1 0.8 1.8 0.0 2.4 0.5 1.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.8 0.5 0.6 0.0 0.9 0.7 0.2 0.0 0.0 0.0 0.0

Gap 37.1 40.0 38.1 41.0 36.8 69.0 45.5 69.1 69.0 44.9 44.2 42.0 39.6 42.0 40.6 40.8 40.0 33.8 45.2 97.1 84.7 47.3 78.7 42.0 70.6 68.0 39.5 53.2 40.5 42.0 41.6 40.2 38.9 39.6 39.4 38.7 38.1 39.6 40.0 38.7 44.3 126.9 70.0 75.3 46.2 41.4 93.8 47.1 41.7 41.0 39.8 38.7

t (s) 1000 1090 1150 1220 1030 1040 1100 1100 1180 1810 1810 1810 1810 1810 1810 1810 1810 601 807 878 941 735 870 927 982 866 870 935 939 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 794 881 947 1019 841 874 951 998 1724 1724 1724 1724

Best

ILS

2.0 5.2 5.0 3.2 3.7 3.7 1.8 3.5 0.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.6 4.3 5.4 3.5 5.5 4.0 2.4 1.3 2.1 4.9 1.9 3.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.1 3.1 2.9 4.3 3.9 3.9 3.6 2.6 0.0 0.0 0.0 0.0

Gap 3.8 1.8 2.5 3.0 1.8 2.1 2.0 3.6 2.5 1.1 1.1 1.0 1.0 1.0 1.0 1.0 0.8 1.4 1.7 2.2 3.8 2.9 3.5 3.3 3.2 2.1 2.0 2.0 3.1 1.3 1.1 0.9 0.6 0.9 0.9 0.8 0.5 1.0 0.9 0.7 1.9 2.3 2.0 1.7 1.5 2.5 1.9 2.0 2.1 1.1 0.9 0.8

t(s) 1020 1150 1210 1246 1060 1080 1120 1126 1182 1810 1810 1810 1810 1810 1810 1810 1810 608 834 924 967 757 905 945 970 885 890 949 957 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 810 903 954 1059 862 900 980 1025 1724 1724 1724 1724

Best

ABC

1020.0 1150.0 1208.0 1240.0 1060.0 1080.0 1120.0 1126.0 1182.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 603.6 828.0 920.2 961.0 753.8 897.6 937.2 967.0 885.0 889.0 948.0 948.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 802.6 902.0 950.0 1059.0 861.8 897.4 977.2 1023.0 1724.0 1724.0 1724.0 1724.0

Avg 0.0 0.0 0.2 1.6 0.9 0.0 0.0 1.2 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.2 1.8 0.8 1.4 3.1 0.9 1.3 2.8 0.0 2.8 0.5 2.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.8 2.6 0.6 1.5 1.3 1.0 0.2 0.0 0.0 0.0 0.0

Gap 13.6 27.3 43.5 38.1 18.6 31.7 23.4 24.0 27.5 1.9 0.9 0.5 0.2 0.9 0.2 0.3 0.2 22.3 19.9 37.0 36.3 19.8 30.9 36.3 46.7 31.1 24.3 34.2 28.8 1.0 0.5 0.2 0.1 0.3 0.2 0.1 0.1 0.2 0.2 0.1 22.5 44.5 37.6 28.5 20.2 34.7 26.0 27.5 2.0 0.7 0.4 0.3

t (s) 1020 1090 1150 1200 1040 1040 1080 1090 1130 1810 1810 1810 1810 1810 1810 1810 1810 611 814 888 925 767 885 908 947 867 843 914 916 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 796 881 948 1021 861 876 945 961 1724 1724 1724 1724

Best

18

1005.0 1083.0 1130.0 1187.0 1031.0 1038.0 1071.0 1077.0 1117.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 605.1 805.7 877.8 914.7 759.0 861.3 897.3 929.6 841.2 835.6 898.3 898.8 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 1458.0 789.2 870.3 921.1 997.0 850.0 854.9 928.1 934.7 1723.3 1724.0 1724.0 1724.0

Avg 1.5 5.8 6.6 5.8 3.6 3.9 4.4 5.5 6.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 4.4 5.4 6.2 2.4 4.9 5.5 6.6 4.9 8.7 5.7 7.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.7 4.3 5.5 6.4 2.9 6.0 6.0 8.8 0.0 0.0 0.0 0.0

Gap

7.4 9.6 12.9 16.5 8.2 8.9 9.5 10.2 11.7 11.5 12.5 14.3 17.1 12.1 13.1 13.7 14.3 5.2 7.6 9.5 11.0 6.8 8.4 9.8 11.1 8.1 8.3 9.5 8.8 12.1 12.6 14.5 16.2 14.5 15.2 16.2 17.2 15.2 14.8 16.7 5.9 7.0 7.9 8.4 6.9 6.7 7.2 7.6 12.3 12.6 14.7 16.7

t (s)

(continued on next page)

GRILS

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

1724 1724 1724 1724

rc205 rc206 rc207 rc208

1724 1724 1724 1724

Best

1724.0 1724.0 1724.0 1724.0

Avg.

1724 1724 1724 1724

Worst

0.0 0.0 0.0 0.0 0.3

Gap

19

BKS

308 404 394 489 595 590 298 463 493 594 353 442 467 567 708 674 362 539 562 667

Prob.

pr01 pr02 pr03 pr04 pr05 pr06 pr07 pr08 pr09 pr10 pr11 pr12 pr13 pr14 pr15 pr16 pr17 pr18 pr19 pr20

308 404 394 489 592 583 298 463 493 594 353 441 457 552 707 650 362 539 490 648

Best

ES

308.0 404.0 392.8 488.2 592.0 576.6 298.0 463.0 493.0 569.2 353.0 441.0 457.0 552.0 688.0 620.4 361.6 537.6 482.6 645.8

Avg.

308 404 388 487 592 563 298 463 493 562 353 441 457 552 682 613 360 532 464 637

Worst

0.0 0.0 0.3 0.2 0.5 2.3 0.0 0.0 0.0 4.2 0.0 0.2 2.1 2.6 2.8 8.0 0.1 0.3 14.1 3.2 2.0

Gap

Table A5 Results for Cordeau's problems for m = 1.

BKS

Prob.

ES

Table A4 (continued)

0.0 103.1 0.6 81.6 68.5 16.3 0.1 0.1 30.5 106.3 3.7 0.6 13.2 6.6 54.2 21.7 53.8 55.5 120.5 7.1 37.2

t (s)

0.0 0.0 0.0 0.0 29.3

t (s)

308 404 388 475 591 586 288 463 493 568 351 440 458 543 649 588 362 535 543 629

Best

CP

1724 1724 1724 1724

Best

CP

0.0 0.0 1.5 2.9 0.7 0.7 3.4 0.0 0.0 4.4 0.6 0.5 1.9 4.2 8.3 12.8 0.0 0.7 3.4 5.7 2.6

Gap

0.0 0.0 0.0 0.0 1.2

Gap

305 394 394 489 594 590 298 454 490 568 353 433 466 521 707 619 360 497 538 588

Best

I3CH

1724 1724 1724 1724

Best

I3CH

1.0 2.5 0.0 0.0 0.2 0.0 0.0 1.9 0.6 4.4 0.0 2.0 0.2 8.1 0.1 8.2 0.6 7.8 4.3 11.8 2.7

Gap

0.0 0.0 0.0 0.0 0.1

Gap

20.8 47.9 72.9 109.3 185.4 189.9 26.5 77.4 137.8 222.2 30.8 59.8 89.5 144.4 248.2 228.6 34.7 99.0 164.6 202.7 119.6

t (s)

0.2 0.2 0.2 0.2 103.4

t (s)

308 404 388 489 587 590 298 463 493 579 330 435 453 549 665 673 359 535 510 662

Best

GVNS

1724 1724 1724 1724

Best

GVNS

307.2 403.6 388.0 475.4 578.0 584.2 297.0 463.0 482.0 564.4 329.0 435.0 452.4 540.6 656.6 643.4 345.6 530.8 507.8 655.0

Avg

1724 1724 1724 1724

Avg

0.0 0.0 1.5 0.0 1.3 0.0 0.0 0.0 0.0 2.5 6.5 1.6 3.0 3.2 6.1 0.1 0.8 0.7 9.3 0.7 1.9

Gap

0.0 0.0 0.0 0.0 0.6

Gap

1.2 3.7 4.1 14.7 20.5 29.0 1.7 3.8 12.3 32.7 1.9 6.5 16.2 32.4 29.4 60.9 5.4 10.4 22.1 57.0 18.3

t (s)

3.3 0.1 0.5 0.2 45.1

t (s)

305 404 394 489 589 575 298 462 482 578 351 430 452 540 666 616 362 539 531 626

Best

SSA

1724 1724 1724 1724

Best

SSA

1.0 0.0 0.0 0.0 1.0 2.5 0.0 0.2 2.2 2.7 0.6 2.7 3.2 4.8 5.9 8.6 0.0 0.0 5.5 6.1 2.4

Gap

0.0 0.0 0.0 0.0 0.3

Gap

8.3 29.1 59.9 106.7 281.7 253.4 15.0 76.0 102.3 189.7 10.3 26.3 49.0 134.3 118.5 558.0 37.6 61.8 152.9 475.3 137.3

t (s)

41.6 39.8 39.7 38.9 49.7

t (s)

304 385 384 447 576 538 291 463 461 539 330 431 450 482 638 559 346 479 499 570

Best

ILS

1724 1724 1724 1724

Best

ILS

1.3 4.7 2.5 8.6 3.2 8.8 2.3 0.0 6.5 9.3 6.5 2.5 3.6 15.0 9.9 17.1 4.4 11.1 11.2 14.5 7.2

Gap

0.0 0.0 0.0 0.0 1.7

Gap

0.5 0.6 1.0 1.9 4.6 2.5 0.4 1.0 1.4 3.6 0.3 0.9 1.9 1.1 5.3 4.1 0.2 0.8 2.7 2.5 1.9

t (s)

2.1 1.0 1.0 0.9 1.7

t(s)

308 392 394 489 589 576 298 463 482 574 353 432 461 565 670 614 359 535 544 608

Best

ABC

1724 1724 1724 1724

Best

ABC

307.0 391.2 394.0 486.0 586.8 573.6 298.0 462.0 481.8 570.4 350.6 432.0 458.1 560.1 666.4 613.4 359.0 535.0 541.0 604.8

Avg

1724.0 1724.0 1724.0 1724.0

Avg

0.3 3.2 0.0 0.6 1.4 2.8 0.0 0.2 2.3 4.0 0.7 2.3 1.9 1.2 5.9 9.0 0.8 0.7 3.7 9.3 2.5

Gap

0.0 0.0 0.0 0.0 0.6

Gap

5.8 4.9 12.6 67.4 91.4 247.4 4.3 15.0 126.3 210.3 1.7 23.5 8.9 76.6 274.6 205.1 6.6 22.1 70.7 346.9 91.1

t (s)

1.2 0.8 0.5 0.3 15.6

t (s)

308 392 392 489 579 553 298 463 483 568 338 436 451 526 667 594 360 511 513 628

Best

GRILS

1724 1724 1724 1724

Best

GRILS

306.7 386.6 387.9 466.1 570.1 538.4 296.6 460.7 468.9 549.3 329.4 429.9 440.2 505.0 655.7 580.0 351.3 485.2 499.9 610.2

Avg

1724.0 1724.0 1724.0 1724.0

Avg

0.4 4.3 1.5 4.7 4.2 8.7 0.5 0.5 4.9 7.5 6.7 2.7 5.7 10.9 7.4 13.9 3.0 10.0 11.0 8.5 5.9

Gap

0.0 0.0 0.0 0.0 2.7

Gap

1.0 2.1 2.7 4.4 7.5 7.0 1.2 3.1 4.6 8.0 1.1 2.5 3.4 5.1 9.8 9.1 1.5 3.7 5.9 9.9 4.7

t (s)

12.6 13.5 14.5 16.5 11.5

t (s)

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

502 715 742 926 1101 1076 566 834 909 1134 566 774 843 1017 1220 1231 652 953 1034 1241

pr01 pr02 pr03 pr04 pr05 pr06 pr07 pr08 pr09 pr10 pr11 pr12 pr13 pr14 pr15 pr16 pr17 pr18 pr19 pr20

502 715 742 913 1090 1070 566 829 907 1128 566 765 845 988 1222 1183 646 945 1019 1207

Best

502.0 713.0 726.6 902.8 1086.4 1050.4 566.0 829.0 904.0 1091.6 564.8 764.4 840.4 981.8 1194.8 1162.6 645.2 933.2 1015.4 1187.2

Avg.

502 711 720 892 1085 1034 566 829 902 1060 564 762 833 973 1158 1150 644 927 1010 1180

Worst

0.0 0.3 2.1 2.5 1.3 2.4 0.0 0.6 0.6 3.7 0.2 1.2 0.3 3.5 2.1 5.6 1.0 2.1 1.8 4.3 1.8

Gap

20

BKS

622 943 1010 1294 1482 1514 744 1139 1282 1573 654 1002 1152 1372 1659 1668 841 1282 1417 1690

Prob.

pr01 pr02 pr03 pr04 pr05 pr06 pr07 pr08 pr09 pr10 pr11 pr12 pr13 pr14 pr15 pr16 pr17 pr18 pr19 pr20

616 941 1010 1269 1478 1505 744 1135 1273 1574 654 990 1147 1339 1654 1629 837 1256 1391 1671

Best

ES

616.0 940.2 1008.6 1266.6 1468.8 1481.4 741.2 1129.2 1269.2 1556.4 654.0 987.4 1143.2 1331.2 1636.8 1606.2 828.2 1243.8 1372.8 1658.6

Avg.

616 940 1006 1259 1461 1439 738 1125 1255 1543 654 985 1142 1324 1592 1582 824 1225 1342 1645

Worst

1.0 0.3 0.1 2.1 0.9 2.2 0.4 0.9 1.0 1.1 0.0 1.5 0.8 3.0 1.3 3.7 1.5 3.0 3.1 1.9 1.5

Gap

Table A7 Results for Cordeau's problems for m = 3.

BKS

Prob.

ES

Table A6 Results for Cordeau's problems for m = 2.

23.7 34.2 56.0 62.9 147.9 169.0 74.5 133.0 111.5 146.6 12.2 95.5 115.5 152.0 106.3 183.0 112.2 161.3 137.2 162.2 109.8

t (s)

0.3 40.6 42.6 132.9 95.3 88.7 4.1 8.6 98.1 43.9 16.0 12.5 73.5 20.0 131.9 130.0 95.7 58.3 146.2 76.8 65.8

t (s)

611 925 998 1197 1423 1373 737 1106 1171 1434 654 971 1080 1299 1508 1521 837 1248 1270 1510

Best

CP

502 699 736 856 1036 1014 561 810 887 1073 559 750 803 955 1120 1093 628 910 967 1114

Best

CP

1.8 1.9 1.2 7.5 4.0 9.3 0.9 2.9 8.7 8.8 0.0 3.1 6.3 5.3 9.1 8.8 0.5 2.7 10.4 10.7 5.2

Gap

0.0 2.2 0.8 7.6 5.9 5.8 0.9 2.9 2.4 5.4 1.2 3.1 4.7 6.1 8.2 11.2 3.7 4.5 6.5 10.2 4.7

Gap

622 936 1010 1286 1481 1501 738 1139 1272 1567 654 997 1145 1315 1654 1609 841 1276 1403 1658

Best

I3CH

502 714 731 917 1101 1040 566 824 878 1117 559 768 832 978 1205 1124 639 937 1003 1155

Best

I3CH

0.0 0.7 0.0 0.6 0.1 0.9 0.8 0.0 0.8 0.4 0.0 0.5 0.6 4.2 0.3 3.5 0.0 0.5 1.0 1.9 0.8

Gap

0.0 0.1 1.5 1.0 0.0 3.3 0.0 1.2 3.4 1.5 1.2 0.8 1.3 3.8 1.2 8.7 2.0 1.7 3.0 6.9 2.1

Gap

132.1 260.6 301.5 442.7 650.3 651.2 260.4 307.0 503.1 731.1 151.7 294.3 378.9 533.7 708.1 818.1 184.3 386.6 604.1 909.7 460.5

t (s)

51.8 127.7 175.6 270.1 410.0 427.6 71.8 184.1 304.9 447.0 71.3 143.6 238.6 337.3 479.1 500.5 117.0 231.0 386.1 541.6 275.8

t (s)

614 940 1003 1294 1480 1512 744 1132 1268 1573 651 979 1106 1369 1609 1668 832 1281 1417 1684

Best

GVNS

502 713 721 908 1082 1076 566 829 901 1124 547 767 829 1009 1219 1224 630 927 1018 1232

Best

GVNS

608.4 937.8 997.4 1279.6 1464 1499.4 736.6 1122.6 1260.6 1563.4 647 975.2 1102.2 1357.4 1594.2 1654.6 822.4 1270.8 1393.6 1677.6

Avg

495.0 706.4 718.4 903.8 1073.6 1070.2 557.8 824.4 883.4 1109.0 539.4 750.8 827.0 999.0 1210.4 1217.8 626.0 914.4 1004.4 1224.4

Avg.

1.3 0.3 0.7 0.0 0.1 0.1 0.0 0.6 1.1 0.0 0.5 2.3 4.0 0.2 3.0 0.0 1.1 0.1 0.0 0.4 0.8

Gap

0.0 0.3 2.8 1.9 1.7 0.0 0.0 0.6 0.9 0.9 3.4 0.9 1.7 0.8 0.1 0.6 3.4 2.7 1.5 0.7 1.2

Gap

3.0 18.6 29.9 106.5 154.9 188.9 11.8 44.8 102.6 198.0 5.9 61.0 77.0 130.9 246.6 413.5 15.2 63.5 216.3 277.4 118.3

t (s)

3.6 12.3 19.2 33.3 94.3 71.0 6.0 20.0 49.7 81.5 3.7 28.0 41.3 117.3 126.6 209.6 9.6 39.5 120.3 128.5 60.8

t (s)

614 939 989 1253 1431 1469 742 1131 1236 1477 654 967 1139 1289 1550 1530 838 1262 1329 1593

Best

SSA

502 712 741 905 1053 1022 566 822 854 1069 566 759 825 922 1155 1094 643 929 1017 1154

Best

SSA

1.3 0.4 2.1 3.2 3.4 3.0 0.3 0.7 3.6 6.1 0.0 3.5 1.1 6.0 6.6 8.3 0.4 1.6 6.2 5.7 3.2

Gap

0.0 0.4 0.1 2.3 4.4 5.0 0.0 1.4 6.1 5.7 0.0 1.9 2.1 9.3 5.3 11.1 1.4 2.5 1.6 7.0 3.4

Gap

18.7 42.7 173.2 168.7 226.6 332.4 24.9 157.1 251.3 574.5 12.8 43.0 80.3 166.3 414.2 697.6 38.0 164.1 220.7 681.3 224.4

t (s)

20.2 37.3 217.7 245.0 249.8 439.3 20.5 75.6 120.7 313.2 17.5 39.0 107.0 111.7 444.8 330.7 20.2 88.2 302.7 554.5 187.8

t (s)

598 899 946 1195 1356 1376 713 1082 1144 1473 632 902 1046 1197 1488 1478 808 1165 1238 1514

Best

ILS

471 660 714 863 1011 997 552 796 867 1004 542 727 757 925 1126 1110 624 877 955 1056

Best

ILS

3.9 4.7 6.3 7.7 8.5 9.1 4.2 5.0 10.8 6.4 3.4 10.0 9.2 12.8 10.3 11.4 3.9 9.1 12.6 10.4 8.0

Gap

6.2 7.7 3.8 6.8 8.2 7.3 2.5 4.6 4.6 11.5 4.2 6.1 10.2 9.0 7.7 9.8 4.3 8.0 7.6 14.9 7.2

Gap

0.4 3.9 3.9 9.0 12.5 19.2 1.0 4.3 10.3 27.9 0.5 1.8 8.2 8.3 14.6 28.2 0.9 6.0 10.2 18.2 9.5

t (s)

0.5 1.2 3.3 4.1 7.1 9.8 1.0 5.1 5.2 10.3 0.7 1.3 2.4 8.1 8.2 11.0 1.3 2.9 5.5 10.7 5.0

t (s)

622 934 991 1258 1438 1462 741 1132 1215 1465 654 973 1122 1288 1592 1534 832 1253 1353 1601

Best

ABC

502 712 742 913 1052 1013 561 810 865 1075 561 757 831 946 1154 1110 652 933 1008 1184

Best

ABC

617.4 930.9 985.4 1256.0 1429.8 1449.0 738.2 1130.2 1208.2 1460.6 652.2 966.0 1111.0 1279.4 1588.4 1521.2 826.6 1241.2 1340.2 1587.8

Avg

502.0 704.6 738.4 905.0 1045.0 1009.8 559.2 806.6 859.2 1067.4 558.2 754.8 831.0 939.8 1145.4 1106.4 652.0 929.0 1004.2 1173.3

Avg

0.7 1.3 2.4 2.9 3.5 4.3 0.8 0.8 5.8 7.1 0.3 3.6 3.6 6.7 4.3 8.8 1.7 3.2 5.4 6.0 3.7

Gap

0.0 1.5 0.5 2.3 5.1 6.2 1.2 3.3 5.5 5.9 1.4 2.5 1.4 7.6 6.1 10.1 0.0 2.5 2.9 5.5 3.6

Gap

6.4 42.6 60.1 147.0 400.2 619.1 7.7 42.5 305.3 658.4 7.1 26.3 62.7 140.9 443.0 700.1 19.6 88.6 310.5 677.6 238.3

t (s)

4.1 20.0 75.7 126.2 220.7 445.7 6.7 74.6 225.3 297.9 2.1 19.4 39.5 120.9 356.1 708.8 16.3 77.3 277.5 600.5 185.8

t (s)

602 894 963 1200 1364 1362 725 1074 1175 1433 628 947 1051 1221 1501 1471 785 1152 1199 1481

Best

GRILS

484 682 718 871 1023 980 560 810 837 1036 547 749 764 929 1115 1069 617 860 893 1124

Best

GRILS

592.8 880.3 948.2 1181.9 1349.9 1341.7 718.6 1066.2 1158.0 1408.3 621.1 929.8 1026.4 1200.7 1474.5 1442.5 773.1 1099.7 1181.4 1451.4

Avg

476.5 673.2 706.9 861.0 1010.7 960.2 554.9 795.5 822.2 1011.9 525.4 717.5 746.0 882.5 1077.9 1047.6 601.1 835.3 871.6 1061.5

Avg

4.7 6.6 6.1 8.7 8.9 11.4 3.4 6.4 9.7 10.5 5.0 7.2 10.9 12.5 11.1 13.5 8.1 14.2 16.6 14.1 9.5

Gap

5.1 5.8 4.7 7.0 8.2 10.8 2.0 4.6 9.5 10.8 7.2 7.3 11.5 13.2 11.6 14.9 7.8 12.4 15.7 14.5 9.2

Gap

3.1 8.2 12.4 19.5 30.0 34.4 5.1 12.8 22.0 36.2 3.4 8.7 13.7 20.9 33.7 38.3 5.5 14.1 25.8 39.2 19.4

t (s)

2.3 5.4 7.7 12.4 20.0 21.0 3.5 8.6 13.7 22.8 2.5 6.1 9.2 13.6 23.1 24.2 3.8 9.5 16.3 26.2 12.6

t (s)

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

BKS

657 1079 1246 1585 1844 1886 876 1385 1619 1943 657 1132 1386 1674 2065 2065 934 1539 1760 2062

Prob.

pr01 pr02 pr03 pr04 pr05 pr06 pr07 pr08 pr09 pr10 pr11 pr12 pr13 pr14 pr15 pr16 pr17 pr18 pr19 pr20

657 1075 1247 1566 1840 1861 872 1392 1603 1936 657 1117 1377 1664 2007 1999 931 1496 1750 2082

Best

ES

657.0 1067.8 1236.0 1549.8 1818.6 1851.4 866.4 1372.6 1585.8 1887.8 657.0 1108.6 1350.4 1638.6 1999.0 1967.6 927.2 1489.4 1723.8 2027.8

Avg.

657 1062 1221 1541 1801 1840 855 1364 1571 1857 657 1100 1336 1617 1992 1926 922 1481 1700 1994

Worst

0.0 1.0 0.8 2.2 1.4 1.8 1.1 0.9 2.1 2.8 0.0 2.1 2.6 2.1 3.2 4.7 0.7 3.2 2.1 1.7 1.8

Gap

Table A8 Results for Cordeau's problems for m = 4.

0.5 85.5 145.0 176.8 138.6 222.2 142.3 102.6 151.9 211.8 0.2 105.4 148.5 245.0 171.7 197.1 227.1 79.8 166.7 261.5 149.0

t (s) 657 1063 1198 1486 1665 1680 871 1309 1495 1784 657 1090 1279 1553 1881 1828 927 1496 1543 1902

Best

CP

0.0 1.5 3.9 6.2 9.7 10.9 0.6 5.5 7.7 8.2 0.0 3.7 7.7 7.2 8.9 11.5 0.7 2.8 12.3 7.8 5.8

Gap 657 1073 1232 1585 1838 1835 872 1377 1604 1943 657 1120 1386 1651 2065 2017 934 1539 1750 2062

Best

I3CH

0.0 0.6 1.1 0.0 0.3 2.7 0.5 0.6 0.9 0.0 0.0 1.1 0.0 1.4 0.0 2.3 0.0 0.0 0.6 0.0 0.6

Gap 0.1 380.6 436.6 603.6 902.9 939.6 228.9 429.9 698.5 1044.4 0.1 477.1 672.0 783.2 1161.7 1183.8 332.8 559.5 919.4 1196.6 647.6

t (s) 657 1070 1215 1531 1827 1855 870 1361 1607 1916 657 1123 1334 1648 1946 2014 921 1518 1723 2019

Best

GVNS

655.8 1064 1210 1526 1811 1840 862.6 1358 1595 1904 657 1110 1325 1637 1933 2000 914.6 1505 1688 2012

Avg 0.0 0.8 2.5 3.4 0.9 1.6 0.7 1.7 0.7 1.4 0.0 0.8 3.8 1.6 5.8 2.5 1.4 1.4 2.1 2.1 1.8

Gap 3.6 25.6 57.2 137.2 208.4 309.9 17.9 67.3 171.1 275.1 5.0 36.7 76.6 243.6 428.3 632.0 14.0 93.0 334.1 463.1 180.0

t (s) 657 1069 1201 1535 1759 1839 871 1358 1565 1853 657 1116 1355 1586 1929 1921 926 1455 1695 1901

Best

SSA

0.0 0.9 3.6 3.2 4.6 2.5 0.6 1.9 3.3 4.6 0.0 1.4 2.2 5.3 6.6 7.0 0.9 5.5 3.7 7.8 3.3

Gap 15.5 44.0 156.1 393.2 321.0 721.0 31.6 118.8 252.5 502.0 13.5 134.2 112.6 192.5 518.6 769.7 30.5 104.8 277.6 685.8 269.8

t (s) 644 1014 1162 1452 1665 1696 840 1267 1460 1782 654 1041 1263 1528 1818 1889 889 1352 1560 1846

Best

ILS

2.0 6.0 6.7 8.4 9.7 10.1 4.1 8.5 9.8 8.3 0.5 8.0 8.9 8.7 12.0 8.5 4.8 12.2 11.4 10.5 8.0

Gap 0.2 2.4 10.5 11.6 19.6 35.4 1.6 6.9 13.8 38.7 0.2 1.9 6.6 16.6 19.5 35.9 1.9 5.7 22.2 26.9 13.9

t (s) 657 1058 1199 1529 1761 1775 859 1349 1543 1848 657 1132 1342 1581 1940 1928 932 1474 1680 1966

Best

ABC

655.8 1053.6 1198.4 1525.0 1755.4 1763.6 855.2 1336.8 1532.8 1842.0 657.0 1130.1 1332.4 1569.2 1937.6 1924.0 926.6 1462.8 1670.8 1958.8

Avg 0.2 2.4 3.8 3.8 4.8 6.5 2.4 3.5 5.3 5.2 0.0 0.2 3.9 6.3 6.2 6.8 0.8 5.0 5.1 5.0 3.8

Gap 3.5 38.0 74.7 199.9 330.7 678.3 18.5 87.1 176.4 725.7 0.4 38.7 70.2 211.5 450.5 685.0 21.3 86.5 285.4 826.3 250.4

t (s) 654 1019 1135 1444 1682 1707 835 1285 1479 1794 657 1066 1250 1517 1869 1822 879 1357 1511 1852

Best

GRILS

652.7 1003.4 1124.4 1431.7 1647.7 1666.7 822.6 1278.1 1454.6 1740.8 654.3 1047.6 1237.5 1489.9 1823.3 1778.2 867.7 1329.1 1478.6 1810.3

Avg 0.7 7.0 9.8 9.7 10.6 11.6 6.1 7.7 10.2 10.4 0.4 7.5 10.7 11.0 11.7 13.9 7.1 13.6 16.0 12.2 9.4

Gap

3.9 10.4 16.4 26.9 41.1 49.4 6.4 17.3 31.2 50.4 4.0 10.9 17.7 28.5 45.7 53.2 6.8 17.9 34.0 53.7 26.3

t (s)

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

21

Opt

1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1724 1724 1724 1724 1724 1724 1724 1724 1724 1724 1724 1724

Prob.

c101 c102 c103 c104 c105 c106 c107 c108 c109 c201 c202 c203 c204 c205 c206 c207 c208 r101 r102 r103 r104 r105 r106 r107 r108 r109 r110 r111 r112 r201 r202 r203 r204 r205 r206 r207 r208 r209 r210 r211 rc101 rc102 rc103 rc104 rc105 rc106 rc107 rc108 rc201 rc202 rc203 rc204

1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1458 1458 1458 1452 1458 1458 1453 1458 1458 1454 1456 1456 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1458 1722 1716 1724 1724 1719 1719 1724 1724 1724 1724 1724 1724

Best

ES

1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1458.0 1458.0 1456.2 1449.4 1457.0 1458.0 1452.8 1457.2 1455.6 1451.4 1452.8 1452.6 1458.0 1457.4 1458.0 1458.0 1458.0 1458.0 1455.4 1458.0 1458.0 1458.0 1457.0 1719.4 1712.8 1723.4 1724.0 1715.2 1718.0 1724.0 1724.0 1724.0 1720.4 1724.0 1724.0

Avg.

1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1458 1458 1455 1445 1453 1458 1452 1456 1449 1448 1450 1447 1458 1455 1458 1458 1458 1458 1453 1458 1458 1458 1455 1713 1708 1721 1724 1709 1714 1724 1724 1724 1714 1724 1724

Worst 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.6 0.1 0.0 0.4 0.1 0.2 0.5 0.4 0.4 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.0 0.0 0.0 0.1 0.3 0.6 0.0 0.0 0.5 0.3 0.0 0.0 0.0 0.2 0.0 0.0

Gap 0.1 3.5 0.0 0.0 0.3 0.4 0.1 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.7 61.9 40.9 138.0 87.5 82.8 87.6 99.4 131.3 66.8 189.9 85.1 0.0 21.9 0.3 43.7 0.0 0.0 90.6 0.4 0.0 0.1 95.5 88.7 80.1 171.9 2.1 150.5 119.8 14.2 45.2 0.1 91.7 0.7 0.0

t (s) 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1458 1458 1458 1439 1458 1458 1444 1446 1455 1443 1458 1440 1458 1458 1458 1452 1458 1458 1443 1458 1458 1458 1451 1724 1718 1724 1724 1719 1716 1724 1719 1724 1724 1724 1724

Best

CP

Table A9 Results for Vansteenwegen problems based on Solomon's instances.

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.3 0.0 0.0 1.0 0.8 0.2 1.0 0.0 1.2 0.0 0.0 0.0 0.4 0.0 0.0 1.0 0.0 0.0 0.0 0.5 0.0 0.3 0.0 0.0 0.3 0.5 0.0 0.3 0.0 0.0 0.0 0.0

Gap 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1458 1458 1458 1454 1458 1458 1458 1456 1458 1454 1458 1456 1458 1458 1458 1458 1458 1458 1456 1458 1458 1458 1449 1724 1724 1724 1724 1724 1724 1724 1724 1724 1719 1724 1724

Best

I3CH

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.0 0.0 0.0 0.1 0.0 0.3 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.3 0.0 0.0

Gap 1.3 3.6 0.7 0.6 2.8 178.9 9.1 230.6 0.7 0.5 0.6 0.6 0.7 0.5 0.6 0.6 0.6 1.2 1.9 958.1 1014.3 44.2 84 1069.5 1543.5 474.2 1281.1 571.8 3488.4 0.7 5.5 0.7 35.7 0.8 0.7 948.8 4.5 0.7 0.8 908.3 54.2 125.3 7.1 8.7 31.2 54.2 12.8 165.6 0.8 1516 0.8 0.8

t (s) 1754.0 1794.0 1810.0 1810.0 1810.0 1806.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1432.2 1441.2 1446.6 1418.2 1441.6 1437.6 1435.0 1441.8 1433.4 1433.4 1430.2 1434.4 1458.0 1450.2 1458.0 1452.6 1458.0 1458.0 1449.8 1458.0 1458.0 1458.0 1452.6 1690.2 1685.0 1709.0 1718.0 1689.8 1690.6 1718.4 1713.0 1724.0 1702.8 1724.0 1724.0

Avg

GVNS

3.1 0.9 0.0 0.0 0.0 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.8 1.2 0.8 2.7 1.1 1.4 1.6 1.1 1.7 1.7 1.9 1.6 0.0 0.5 0.0 0.4 0.0 0.0 0.6 0.0 0.0 0.0 0.4 2.0 2.3 0.9 0.3 2.0 1.9 0.3 0.6 0.0 1.2 0.0 0.0

Gap 27 23.2 5.7 1.6 1.6 9.1 1.1 0.7 0.1 0.1 1.8 1.5 0.5 0.1 0.1 0.1 0.1 23.4 20.7 23.5 55 32.3 25.3 46 56.4 42.8 50.1 40 58.7 1.4 19.8 2.8 6.6 0.4 0.7 12.4 0.9 0.2 0.9 14.6 28.1 30.1 37.9 43.6 32.9 40.5 51.5 51.1 1.9 10.3 2.2 0.1

t (s) 1770 1810 1810 1810 1780 1800 1760 1770 1810 1810 1810 1810 1810 1810 1810 1810 1810 1455 1458 1455 1442 1458 1458 1452 1447 1453 1454 1444 1446 1458 1458 1458 1447 1458 1458 1452 1457 1458 1458 1451 1712 1718 1724 1719 1716 1714 1722 1719 1724 1714 1724 1724

Best

SSA

2.2 0.0 0.0 0.0 1.7 0.6 2.8 2.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.0 0.2 1.1 0.0 0.0 0.4 0.8 0.3 0.3 1.0 0.8 0.0 0.0 0.0 0.8 0.0 0.0 0.4 0.1 0.0 0.0 0.5 0.7 0.3 0.0 0.3 0.5 0.6 0.1 0.3 0.0 0.6 0.0 0.0

Gap 89.5 52.9 52.9 44.4 152.7 123.1 61.1 59.6 63 45 44.3 42.1 39.8 42 40.7 40.9 40.1 67.9 84.9 59.6 131.3 136.5 116.3 78.8 87.1 128.1 172.9 114.6 78 42.2 58.5 39.4 70.2 38.7 38.4 155.9 54.2 38.8 39 66.2 79.9 103.4 82.6 87.7 87 63.7 51.4 121 41.7 47.8 38.7 37.9

t (s) 1720 1790 1810 1810 1770 1750 1790 1810 1810 1810 1810 1810 1810 1810 1810 1810 1810 1441 1450 1450 1402 1435 1441 1431 1430 1432 1419 1410 1418 1458 1443 1458 1440 1458 1458 1428 1458 1458 1458 1422 1686 1659 1689 1719 1691 1665 1701 1698 1724 1686 1724 1724

Best

ILS

5.0 1.1 0.0 0.0 2.2 3.3 1.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.2 0.5 0.5 3.8 1.6 1.2 1.9 1.9 1.8 2.7 3.3 2.7 0.0 1.0 0.0 1.2 0.0 0.0 2.1 0.0 0.0 0.0 2.5 2.2 3.8 2.0 0.3 1.9 3.4 1.3 1.5 0.0 2.2 0.0 0.0

Gap 4.1 4.2 3 1.8 2.8 3.8 3.1 2.5 2 1.1 1.1 1 1 1 1 1 0.8 2.5 3.1 2 2.3 4.1 3.1 3.3 2.7 2.5 4.4 3 2.4 1.3 2.7 1.6 3.4 1.1 1.1 1.7 1.6 1.1 1.2 1.9 4.3 2.9 3.4 3.2 3.9 4.8 2.4 5.6 2.1 1.7 2.9 1

t (s) 1794 1810 1810 1810 1791 1789 1789 1797 1810 1810 1810 1810 1810 1810 1810 1810 1810 1458 1458 1458 1440 1458 1458 1458 1455 1458 1451 1452 1448 1458 1458 1458 1458 1458 1458 1454 1458 1458 1458 1456 1717 1713 1724 1724 1724 1723 1724 1724 1724 1719 1724 1724

Best

ABC

22

1.3 0.0 0.0 0.0 1.5 1.3 1.4 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 0.2 1.4 0.0 0.0 0.4 0.4 0.6 0.6 0.6 0.7 0.0 0.0 0.0 0.3 0.0 0.0 0.7 0.0 0.0 0.0 0.5 0.6 1.1 0.2 0.0 0.5 1.0 0.0 0.2 0.0 0.4 0.0 0.0

Gap

59.1 14.3 6.4 1.9 58.8 53.2 53.2 44.2 24.1 1.9 0.9 0.5 0.2 0.9 0.2 0.3 0.2 25.1 18.4 14.3 59 43.2 32.4 45.8 64.9 64.3 54.8 47.5 43.1 1 5.8 0.8 35.6 1.3 0.6 35.7 12.6 0.9 0.6 29.3 52.9 38.8 40.9 43.7 47.6 50.5 46 47.5 2 12.7 2.1 1.1

t (s)

(continued on next page)

1786.0 1810.0 1810.0 1810.0 1782.0 1786.0 1784.0 1792.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1810.0 1457.4 1455.4 1455.0 1437.8 1458.0 1458.0 1452.0 1451.8 1449.4 1449.0 1449.8 1448.0 1458.0 1458.0 1458.0 1453.4 1458.0 1458.0 1448.4 1457.8 1458.0 1458.0 1451.2 1713.6 1705.2 1721.0 1723.4 1716.0 1706.0 1724.0 1720.6 1724.0 1717.4 1724.0 1724.0

Avg

K. Karabulut and M.F. Tasgetiren

Computers & Industrial Engineering 139 (2020) 106109

1724 1724 1724 1724

rc205 rc206 rc207 rc208

1724 1724 1724 1724

Best

1724.0 1724.0 1724.0 1724.0

Avg.

1724 1724 1724 1724

Worst 0.0 0.0 0.0 0.0 0.1

Gap 0.0 0.6 0.5 0.0 37.5

t (s) 1724 1724 1724 1724

Best

CP

23

BKS

657 1220 1788 2477 3351 3671 948 2006 2736 3850

Prob.

pr01 pr02 pr03 pr04 pr05 pr06 pr07 pr08 pr09 pr10

619 1209 1774 2477 3351 3671 940 2006 2736 3850

Best

ES

615.8 1206.0 1770.0 2473.8 3351.0 3670.8 938.2 2006.0 2736.0 3850.0

Avg.

614 1199 1766 2470 3351 3670 937 2006 2736 3850

Worst

6.3 1.1 1.0 0.1 0.0 0.0 1.0 0.0 0.0 0.0 1.0

Gap 102.0 155.2 94.5 83.1 45.1 143.3 100.4 41.9 11.6 6.3 78.3

t (s) 622 1198 1775 2468 3351 3671 942 2006 2736 3850

Best

CP

Table A10 Results for Vansteenwegen problems based on Cordeau's instances.

Opt

Prob.

ES

Table A9 (continued)

5.3 1.8 0.7 0.4 0.0 0.0 0.6 0.0 0.0 0.0 0.9

Gap

0.0 0.0 0.0 0.0 0.2

Gap

619 1207 1781 2477 3351 3671 943 2006 2736 3850

Best

I3CH

1724 1724 1724 1724

Best

I3CH

5.8 1.1 0.4 0.0 0.0 0.0 0.5 0.0 0.0 0.0 0.8

Gap

0.0 0.0 0.0 0.0 0.0

Gap

146.7 669.7 1383.7 641.9 19.3 30.0 299.7 55.9 10.8 9.1 326.7

t (s)

0.7 0.8 0.8 0.9 265.2

t (s)

608.4 1198.8 1760.8 2467.4 3351.0 3670.6 935.0 2004.6 2736.0 3850.0

Avg

GVNS

1723.0 1724.0 1724.0 1724.0

Avg

GVNS

7.4 1.7 1.5 0.4 0.0 0.0 1.4 0.1 0.0 0.0 1.2

Gap

0.1 0.0 0.0 0.0 0.6

Gap

3.1 27.9 50.2 114.0 43.2 196.4 13.8 46.9 8.5 9.4 51.3

t (s)

3.2 2.5 2.2 0.2 16.9

t (s)

614 1205 1758 2461 3351 3661 948 2006 2736 3847

Best

SSA

1724 1724 1724 1724

Best

SSA

6.5 1.2 1.7 0.6 0.0 0.3 0.0 0.0 0.0 0.1 1.0

Gap

0.0 0.0 0.0 0.0 0.3

Gap

18.5 48.0 124.2 393.3 1109.4 1646.7 27.2 128.5 794.8 1369.2 566.0

t (s)

41.4 45.4 39.7 38.4 70.3

t (s)

608 1180 1738 2428 3297 3650 909 1984 2729 3850

Best

ILS

1724 1708 1713 1724

Best

ILS

7.5 3.3 2.8 2.0 1.6 0.6 4.1 1.1 0.3 0.0 2.3

Gap

0.0 0.9 0.6 0.0 1.1

Gap

0.7 4.5 11.3 45.4 37.3 106.1 1.5 12.0 33.0 52.3 30.4

t (s)

2.1 1.3 1.5 1.1 2.4

t (s)

623 1215 1769 2477 3351 3671 936 2006 2736 3850

Best

ABC

Missing 1724 1724 1724

Best

ABC

617.4 1203.8 1764.6 2474.2 3351.0 3670.6 931.4 2006.0 2736.0 3850.0

Avg

1724.0 1724.0 1724.0

Avg

6.0 1.3 1.3 0.1 0.0 0.0 1.8 0.0 0.0 0.0 1.1

Gap

0.0 0.0 0.0 0.3

Gap

6.4 32.3 138.6 266.7 331.1 752.0 14.9 37.7 104.8 126.8 181.1

t (s)

4.3 1.9 1.1 24.6

t (s)

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K. Karabulut and M.F. Tasgetiren

Appendix B. New best-known solutions Solomon's problems for m = 2 r209 Profit: 1417 Tour1: 27 69 63 64 11 62 88 31 52 47 19 82 83 5 98 44 14 38 86 16 61 85 99 94 87 57 22 40 53 26 3 78 34 50 20 70 1 35 66 32 48 17 60 89 13 58 Tour2: 95 59 42 15 2 23 67 39 75 72 21 28 12 76 29 79 33 9 71 65 81 51 30 10 49 36 8 45 84 93 96 97 37 100 43 54 4 74 56 25 55 24 80 68 77 Cordeau's problems for m = 2 pr13 Profit: 845 Tour1: 109 107 87 14 80 26 139 110 27 31 44 124 2 94 48 130 23 81 90 7 32 86 Tour2: 138 66 53 82 29 33 128 25 22 123 18 16 104 43 63 5 28 35 39 17 91 129 10 112 79 70 74 141 pr15 Profit: 1222 Tour1: 150 11 235 29 165 153 54 18 238 67 71 164 176 40 216 47 147 2 215 27 26 179 237 43 204 123 85 182 136 62 77 198 64 212 79 32 170 118 5 Tour2: 146 145 3 107 205 112 185 74 24 66 181 230 28 135 197 80 187 203 142 81 217 102 129 157 193 84 21 48 168 Cordeau's problems for m = 3 pr10 Profit: 1574 Tour1: 261 143 166 108 263 206 64 50 229 280 138 189 238 281 65 77 9 260 112 160 233 207 144 173 177 17 129 51 161 69 Tour2: 178 170 79 185 109 61 25 219 284 114 97 267 181 232 205 249 35 86 126 100 179 118 62 14 54 184 279 116 Tour3: 39 196 11 220 191 142 87 275 20 34 42 2 115 158 243 27 72 155 242 71 101 59 285 156 208 6 110 128 152 Cordeau's problems for m = 4 pr03 Profit: 1247 Tour1: 118 39 17 87 139 88 52 80 129 91 26 71 35 107 109 78 14 10 112 12 Tour2: 115 101 105 37 51 58 68 6 117 56 24 70 79 135 125 61 111 34 19 4 25 Tour3: 130 122 15 28 5 104 86 65 29 53 82 59 93 33 128 123 18 22 43 63 144 Tour4: 7 64 38 99 23 81 32 138 96 90 134 2 124 44 94 48 142 77 pr08 Profit: 1392 Tour1: 74 72 43 20 98 17 75 44 54 135 56 24 65 59 123 13 47 141 101 Tour2: 63 129 84 93 114 115 127 97 106 133 34 109 10 128 113 8 16 23 28 140 19 Tour3: 136 30 100 83 36 52 58 48 131 55 92 107 21 85 1 33 41 144 71 111 Tour4: 117 87 37 108 68 77 125 32 76 64 112 14 6 70 67 124 60 138 pr20 Profit: 2082 Tour1: 92 18 192 144 260 9 112 160 118 179 100 108 122 244 98 62 246 80 64 50 206 14 54 65 184 77 103 138 123 263 233 251 177 190 143 207 Tour2: 161 146 51 35 86 126 249 180 119 267 97 25 114 219 284 124 129 277 17 22 166 32 168 228 227 273 271 Tour3: 7 155 44 243 27 72 158 185 115 109 221 181 232 205 49 20 134 34 42 141 275 87 94 69 43 2 258 6 16 71 110 Tour4: 46 21 1 208 222 242 56 101 150 105 89 174 285 196 245 11 39 191 91 178 142 36 154 52 96 254

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