Aghezzaf and Fahim SpringerPlus (2016)5:1781 DOI 10.1186/s40064-016-3440-6

Open Access

RESEARCH

Iterated local search algorithm for solving the orienteering problem with soft time windows Brahim Aghezzaf and Hassan El Fahim* *Correspondence: [email protected] Laboratoire Informatique et Aide à la Décision (LIAD), Département de Mathématiques et Informatique, Faculté des Sciences Aïn Chock, Université Hassan II de Casablanca, Km 8 Route d’El Jadida, 5366 Maarif, 20100 Casablanca, Morocco

Abstract  In this paper we study the orienteering problem with time windows (OPTW) and the impact of relaxing the time windows on the profit collected by the vehicle. The way of relaxing time windows adopted in the orienteering problem with soft time windows (OPSTW) that we study in this research is a late service relaxation that allows linearly penalized late services to customers. We solve this problem heuristically by considering a hybrid iterated local search. The results of the computational study show that the proposed approach is able to achieve promising solutions on the OPTW test instances available in the literature, one new best solution is found. On the newly generated test instances of the OPSTW, the results show that the profit collected by the OPSTW is better than the profit collected by the OPTW. Keywords:  Combinatorial optimization, Orienteering problem, Soft time window, Iterated local search, Variable neighborhood search

Background In orienteering problem (OP) a set of potential customers is given; the service for these customers is optional during the current planning time horizon since the travel cost of the route is limited. The travel cost is often expressed as the travel time or the travel distance. Thus, a positive value called profit is associated with every customer making its visit more or less attractive. The name of this routing problem originates from a game in which competitors have to visit a set of control points in a given area. If the control point is visited, the competitor gets a profit. The winner of the game is the competitor who collects maximum profits and gets to the end point within a prescribed amount of time. As a routing problem, the OP consists in finding the route visiting a subset of customers that maximizes the total collected profit while satisfying the maximum duration constraint. The OP is also known in the literature as the Selective Traveling Salesman Problem (Thomadsen and Stidsen 2003), the Maximum Collection Problem (Butt and Cavalier 1994) and the Bank Robber Problem (Awerbuch et al. 1998). Few vehicle routing problems have such applicability as OP. This problem arises in a variety of applications including design of tourist trips to maximize the value of the visited attractions (Vansteenwegen and Oudheusden 2007), recruiting of athletes from high schools for a college team (Butt and Cavalier 1994), delivery of home heating fuel where © 2016 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Aghezzaf and Fahim SpringerPlus (2016)5:1781

the urgency of a customer for fuel is treated as a profit (Golden et al. 1984), routing of oil tankers to serve ships at different locations (Golden et al. 1987) and reverse logistics problem of a firm that aims to collect used products from its dealers (Aras et al. 2011). The OP is a well-studied combinatorial optimization problem that was first presented and heuristically solved by Tsiligirides (1984). Several heuristics and metaheuristics were proposed for the solution of the OP [the reader is referred for example to the papers by Tasgetiren (2001), Ramesh and Brown (1991) and Gendreau et al. (1998)]. In the time windows version of the OP called the orienteering problem with time windows (OPTW), customers have hard time windows and service times. In hard time windows, arriving at customer later than latest time of its time window is strictly forbidden. A waiting is incurred if the vehicle reaches to a customer before its earliest time window. In OPTW, the objective is designing the route that maximizes the total collected profit while satisfying the time limit duration and the hard time windows constraints. In recent years there has been considerable interest for the OPTW which has led to a significant body of literature. The authors in Righini and Salani (2009) proposed a bidirectional dynamic programming algorithm for solving the OPTW to optimality. They use a technique named decremental state space relaxation in which the dynamic programming algorithm takes advantage of a state space relaxation. The authors in Duque et  al. (2015) proposed an exact algorithm based on pulse framework for solving the OPTW to optimality. Most studies have focused on designing heuristic algorithms, several heuristics and metaheuristics were then proposed for the solution of the OPTW [the reader is referred for example to the papers by Vansteenwegen et al. (2009), Lin and Yu (2012), Labadie et al. (2011, 2012), Montemanni and Gambardella (2009), Tunchan (2014), Gunawan et al. (2015) and Lahyani et al. (2016)]. For a recent survey on OP and OPTW the reader is referred to the paper by Gunawan et al. (2016). Problem related to the problem studied in this research is the vehicle routing problem with soft time windows (VRPSTW). In many real-life problems, some or all customers’ time windows are not so strict that can be violated with appropriate penalties. Such kind of time constraint is called soft time window. In VRPSTW, vehicles are allowed to serve customers before the earliest and/or after the latest time windows bounds. This type of time windows is useful for the dispatcher when: ••  The number of routes needed for hard time windows exceeds the number of available vehicles. ••  A study of cost-service tradeoffs is required. ••  The dispatcher has qualitative information regarding the relative importance of hard time windows across customers. Besides, relaxing time windows can result in lower total costs without hurting customers’ satisfaction significantly. In the literature, there are different ways of relaxing time windows which lead to different variants of VRPSTW. ••  If a vehicle arrives before the earliest bound of the time window, it waits as in hard time windows case. However, late service is allowed if an appropriate penalty is paid. The authors in Taillard et al. (1997) proposed a tabu search heuristic for this variant.

Page 2 of 36

Aghezzaf and Fahim SpringerPlus (2016)5:1781

••  Both early and late services are allowed by paying appropriate penalties. The authors in Koskosidis and Solomon (1992) proposed an optimization-based heuristic for this variant. ••  Both early and late services are allowed as in the second variant. However, the maximum allowable violation of the time windows and the maximum waiting time allowed are limited. This is the variant studied by Chiang and Russell (2004) and Balakrishnan (1993). The authors in Balakrishnan (1993) described three heuristics for solving this variant. While the authors in Chiang and Russell (2004) proposed a tabu search heuristic to deal with this variant. For more detail about these relaxation schemes and the algorithms proposed in the literature to solving them, the reader is referred to the paper by Vidal et al. (2015). Contributions

We observed when solving orienteering problems with time windows as in Aghezzaf and Fahim (2015) that the gap between the total travel time of a route and the travel time limit is significant especially on instances with long scheduling horizon. Thus, we have decided to manage this gap by allowing relaxation of time windows in order to improve the profit collected by the vehicle. Furthermore, there are many practical reasons for allowing violation of time windows: ••  Many applications do not require hard time windows. ••  In many cases travel times cannot be accurately known. ••  Customers may be unwilling to set precise time windows in advance and simply prefer the flexibility to alter their delivery requests. The contribution of this paper is twofold: ••  We introduce and define a new routing denoted the orienteering problem with soft time windows (OPSTW). To the best of our knowledge, this is the first study dealing with orienteering problem with soft time windows. In this routing problem, late service is allowed if an appropriate penalty is paid. In this relaxation scheme, we are placing restrictions on both the penalty payable and the waiting time. We think that OPSTW solutions can result in routes visiting a significant number of potential customers without hurting customers’ satisfaction significantly. Furthermore, soft time windows can provide a workable plan of action for decision makers when hard time windows are not required or when it is not possible to visit all customers during the current planning time horizon which is the case for the OPTW. ••  We develop a hybrid algorithm that combines an iterated local search with a variable neighborhood search for this specific problem. We also apply it to standard instances and compare its performance to that of other algorithms proposed in the literature for the OPTW. The rest of this paper includes four additional sections. “Mathematical model” section defines the mathematical notation and formulation of OPSTW. “Solution algorithm” section describes the proposed hybrid algorithm. “Computational results” section presents

Page 3 of 36

Aghezzaf and Fahim SpringerPlus (2016)5:1781

Page 4 of 36

the computational results and compares our algorithm against published results both with regard to solution quality and computational time. The last section is devoted to the conclusions.

Mathematical model The OPTW studied in this paper can be described as follows: let G = (V , E) be a complete graph, where V = {0, 1, . . . , n} is a vertex set and E = {(i, j) ∈ V 2 i � = j} is an arc set. Vertex 0 denotes a depot at which the vehicle starts and ends its tour. The set of vertices C = {1, . . . , n} specify the location of a set of n customers. Each vertex i ∈ V has an associated profit pi ( p0 = 0), a service time Si (S0 = 0) and a time window [ei , li ] which is assumed to be hard. Each arc (i, j) ∈ E has an associated cost tij which represents the time required to travel from vertex i to vertex j. The cost tij is defined as the Euclidean distance between the points corresponding to i and j. The arrival time to a customer i is denoted ai; the beginning of service time is denoted bi. The objective is to design a route R that maximizes the total collected profit subject to the following: ••  The route R cannot start before e0 and cannot end after l0. ••  The service to a customer i cannot start before ei and if the vehicle arrives too early it can wait for a certain period of time wi = max(ei − ai , 0) and serves that customer. ••  Every customer is visited at most once. ••  The total travel time of R is limited by a time limit Tmax. In order to formulate the model we define the following decision variables:

xij binary variable equal to 1 if the vehicle travels directly from vertex i to vertex j, and 0 otherwise. yi binary variable equal to 1 if vertex i is visited, and 0 otherwise. bi beginning of service time at customer i. M is a large value. The OPTW can be formulated as the following mixed integer linear programming model:  max f = pi yi (1) i∈C

subject to:   x0j = xi0 = 1 j∈C



xil =

i∈V



xlj ≤ 1 ∀l ∈ C

i∈V

(3)

j∈V

bi + Si + tij − bj ≤ M(1 − xij ) �

(2)

i∈C



Si yi +

� j∈V

∀i, j ∈ V

(4)



tij xij  ≤ Tmax

(5)

Aghezzaf and Fahim SpringerPlus (2016)5:1781

ei ≤ bi ≤ li

Page 5 of 36

∀i ∈ V

(6)

xij , yi ∈ {0, 1} ∀i, j ∈ V

(7)

bi ∈ R+ ∩ [e0 , l0 ]

(8)

∀i ∈ V

The objective function (1) maximizes the total collected profit. Constraint (2) guarantees that the route starts and ends at vertex 0 (depot). Constraints (3) and (4) determine the connectivity and the time line of the route. Constraint (5) ensures the maximum time duration constraint of the route. Constraints (6) restrict the start of the visit to the time windows. Constraints (7) and (8) are variables definition. In OPSTW, the time window of every customer i ∈ C can be enlarged to an outer time window [ei , li + Pmax ] = [ei , ˆli ], where Pmax is an upper bound on the maximum allowpenalty able time window violation. An appropriate penalty Pi is then paid if the service ˆ starts late that is ai ∈]li , li ]. The penalty function can be defined as follows: penalty Pi

=



0

if ei − Wmax ≤ ai ≤ li

ai − li

if li < ai ≤ ˆli

(9)

One can express the OPSTW objective function as a combination between the total collected profit (the classic objective for OPTW) and the total penalty for time windows violations. In our formulation, we do not express it that way since this will change the nature of the faced problem and the aim of this work. In our OPSTW formulation, the total penalty and the total waiting time are expressed as travel costs and added to the total travel time of the route. Then, constraint (5) of the previous model changes as follows:   � � � �  Si + P penalty + wi yi + tij xij  ≤ Tmax (10) i j∈V

i∈V

Since in our formulation we take into account the fact that wi ≤ Wmax for each routed customer i, the following constraint is added to the model.

(ej − (bi + Si + tij ))xij ≤ (Wmax )xij

∀i, j ∈ V

(11)

Regarding the time windows constraints they change as follows:

ei ≤ bi

∀i ∈ V

(13)

b0 ≤ l0 bi ≤ ˆli

(12)

∀i ∈ C

(14)

In the next subsection, we will describe the approach that we propose to deal with the OPSTW.

Aghezzaf and Fahim SpringerPlus (2016)5:1781

Solution algorithm At its core, the approach proposed to solve the OPSTW is an iterated local search (ILS). ILS is a local search based metaheuristic that was introduced in Lourenço et al. (2003) to solve combinatorial optimization problems. Let S be the starting solution for the ILS process. At each iteration, a diversification phase is firstly applied by perturbing S. An intensification phase is then performed around the perturbation output by applying a local search procedure to produce a solution S´ . If S´ satisfies an acceptance criterion, it replaces the starting solution and the next perturbation phase is performed from that solution. Otherwise, S´ is discarded and the search returns to the previous starting solution. In the proposed ILS algorithm, a variable neighborhood search (VNS) is applied to S´ even if it is better than S or not. The objective is intensifying the search around S´  , which is a local optima with respect to the local search procedure, in order to explore promising regions of the solution space. Algorithm 1 illustrates the steps of the proposed hybrid ILS (HILS) algorithm. The stopping condition used is the maximum number of iterations allowed Lmax. In the next subsections, we will describe the components of the proposed ILS algorithm which are the initial solution procedure, the perturbation operator, the local search procedure and the variable neighborhood search.

Initial solution

We propose three insertion heuristics to generate a set of solutions and we pick the best as the starting solution for the ILS process. These heuristics follow the scheme of the insertion heuristic proposed by Solomon (1987) for the vehicle routing problem with time windows (VRPTW), they differ in the expression of the criterion used to compute the best feasible insertion place of each unrouted customer on the current partial route. In the following subsections we will first present the procedure implemented to check the feasibility of an insertion and then we will present the initial solution procedure. Feasibility check

Let R = (i0 , i1 , i2 , . . . , im−1 , im ), i0 = im = 0 be the current partial route and let u be an unrouted customer. We define a feasible place of the customer u in R as a position

Page 6 of 36

Aghezzaf and Fahim SpringerPlus (2016)5:1781

Page 7 of 36

(ip−1 , ip ) p ∈ {1, . . . , m} in R for which, if u is inserted between the adjacent vertices ip−1 and ip then: The waiting time at customer u given by Eq. (15) satisfies the following: wu = max(0, eu − (bip−1 + Sip−1 + tip−1 u )) ≤ Wmax

(15)

The time duration constraint on the route is still satisfied, that is: p−1 

tir−1 ir +

r=1

+

p−1  

penalty

wir + Sir + Pir



penalty

+ tip−1 u + tuip − tip−1 ip + Su + wu + Pu

r=1 m 

tir−1 ir +

m−1 

penalty

wir + Sir + Pir



≤ Tmax

(16)

r=p

r=p+1

All vertices subsequent to the inserted customer u still satisfy at most their outer time windows. This is done using the conditions for time feasibility proposed by Solomon (1987) as follows:

bu ≤ lˆu ,

and bir + PFir ≤ lˆir ,

p≤r≤m

(17)

where:

− bip PFip = binew p

(18)

corresponds to the beginning of service at customer ip (which is the The metric binew p arrival time at the depot if p = m) given that customer u is inserted between ip−1 and ip. This metric is computed as follows:

binew = max(eip , bu + Su + tuip ) p

(19)

It is clear that PFip ≥ 0 since the matrix (tij )(i,j)∈E satisfies the triangle inequality. This metric is computed for the rest of the subsequent vertices as follows:

PFir+1 = max{0, PFir − wir+1 },

p≤r ≤m−1

(20)

Insertion heuristics

Each insertion heuristic Hl, l ∈ {1, 2, 3} starts by determining the best feasible place of each unrouted customer on R. Such position is defined as the position (i(u),  j(u)) for which: H

C1 (i(u), u, j(u)) = min[C1 l (ip−1 , u, ip )],

p ∈ {1, 2, . . . , m}

(21)

The first insertion heuristic H1 computes this position using the following criterion:

− bip ) C1H1 (ip−1 , u, ip ) = α1 (tip−1 u + tuip − tip−1 ip + α3 Su ) + α2 (binew p

(22)

The second insertion heuristic H2 computes this position using the following criterion:

C1H2 (ip−1 , u, ip ) = α1 (tip−1 u + tuip − tip−1 ip + α3 Su ) + α2 wu

(23)

Aghezzaf and Fahim SpringerPlus (2016)5:1781

Page 8 of 36

While the third insertion heuristic H3 computes this position using the following criterion: H

C1 3 (ip−1 , u, ip ) = α1 (tip−1 u + tuip − tip−1 ip + α3 Su ) + α2 θu

(24)

The metric θu corresponds to the time difference between the completion of service at customer ip−1 (which is the departure time from the depot if p = 1) and the beginning of service at customer u. This metric is expressed as follows:

θu = bu − (bip−1 + Sip−1 )

(25)

The weights α1, α2 and α3 define the relative contribution of each individual metric in the computing of the best feasible insertion place. The parameter α1 takes into account the saving in travel time by inserting u between ip−1 and ip. Then, each insertion heuristic selects the best unrouted customer v according to the criterion given by Eq. (26) and inserts it in the current partial route R.

C2 (i(v), v, j(v)) = min[C2 (i(u), u, j(u))] u∈∇

(26)

where ∇ is the set of unrouted customers having at least one feasible place on the current partial route. The criterion C2 (i(u), u, j(u)) is expressed as follows: H

C2 (i(u), u, j(u)) =

C1 l (i(u), u, j(u)) puα4

(27)

The parameter α4 is the exponent of the profit of customer u. The procedure of customer insertion is repeated, for each insertion heuristic, until no further unrouted customer can be inserted into R. The insertion procedure terminates by providing the set of assigned customers and the sequence in which these customers are routed. This is repeated for a number of values for αi , i ∈ {1, 2, 3, 4} and the best overall solution is returned at the end. We compare two solutions using the following criteria in decreasing order: collected profit and time duration. Note that αi , i ∈ {1, 2, 3, 4} are positive weights that satisfy: αi ≥ 0, i ∈ {1, 2, 3, 4} and α1 + α2 = 1. Perturbation

The perturbation operator used in the proposed ILS algorithm performs, around a solution, by selecting the customer with minimal profit and removing it from this solution. Note that after a removal, all vertices (customers + depot) following removed one are shifted towards the beginning of the route in order to ensure its continuity. As one can intuitively expect, this forward move can result in an infeasible solution since the waiting time on some customers may exceed the maximum allowable waiting time Wmax. This is solved using local search procedure that will be described in the next subsection.

Aghezzaf and Fahim SpringerPlus (2016)5:1781

Local search

The local search procedure tries to fulfill the available room in the solution, obtained through perturbation, by inserting other feasible unrouted customers. On one hand and as one can intuitively expect, evaluating the possible insertion of each unrouted customer using the criterion presented in the initial solution procedure will increase the risk of inserting the set of customers just removed, and getting easily stuck in the initial solution. On the other hand, the solution obtained through perturbation can be infeasible. Given a starting solution S, the solution obtained through perturbation is denoted Sˆ . The local-search procedure performs, around Sˆ , by inserting each unrouted customer in its first feasible place. Using this local search procedure, following effects are observed: ••  ••  ••  •• 

an unrouted customer in S can be part of Sˆ . a routed customer in S cannot be inserted in Sˆ . the position of a customer in S can be changed to another position in Sˆ . the customers of Sˆ are shifted towards the end of the route in order to avoid unnecessary waiting time. As a consequence, the solution obtained after local search procedure is feasible.

This procedure will help to re-optimize a solution, make it feasible if it is not and insert other feasible unrouted customers in order to improve the value of the incumbent solution. Variable neighborhood search

Variable neighborhood search (VNS) is a local search based metaheuristic that was introduced in Mladenović and Hansen (1997) for solving combinatorial optimization problems. The basic idea behind this metaheuristic is a systematic change of neighborhoods both in descent phase using a local search procedure, and in shaking phase using a set of neighborhood structures. VNS is a stochastic algorithm in which, first, a finite set of neighborhood structures is defined. We denote by N(S) the set of solutions in the neighborhood of S. Each VNS iteration is composed of three steps: shaking, local search and move. At each iteration, a solution S´ is randomly generated from the neighborhood of S. A local search is then applied with S´ as the initial solution, the obtained solution is denoted Sˆ . If Sˆ is better than S, the search moves to Sˆ and continues by considering the first neighborhood structure. Otherwise, k is incremented. The steps of the proposed VNS algorithm are given by Algorithm 2. Before defining the neighborhood structures used in the developed VNS, we first define two metrics ϕS , ϑS ∈ N. To do that, let us denote with S = (0, c1 , c2 , . . . , cm , 0) ch ∈ C, h ∈ {1, . . . , m} a feasible solution. These metrics are defined as follows:

Page 9 of 36

Aghezzaf and Fahim SpringerPlus (2016)5:1781

Page 10 of 36

ϕS =

m

(28)

ϑS =

m

(29)

2

2

The first neighborhood of S denoted N1 (S) is defined as the set of all solutions obtained by removing two customers, one from the set (c1 , c2 , . . . , cϕS ) and the other from the set (cϑS , cϑS +1 , . . . , cm ). The second neighborhood N2 (S) is defined as the set of all solutions obtained by removing a set of customers (ck : k ∈ [1, j]) such that j ∈ [1, m]. The idea behind using floor and ceiling functions to define these neighborhood structures is to reduce the size of the neighborhoods and then the computational time.

Computational results The proposed algorithm is coded in Java; the experiments are performed on a personal computer Intel(R) with 2.1 GHz and 4 GB of RAM. We test, first, our algorithm on OPTW test instances. Based on these instances, we design test instances for the OPSTW. Each experiment is performed, on each test instance, five times for which the average results are presented. The presented computational times are the average times over five runs. Test instances

The authors in Righini and Salani (2009) designed test instances for OPTW based on 29 VRPTW test instances of Solomon (1987) namely C100, R100 and RC100, and on 10 Multi-depot Vehicle Routing Problem (MDVRP) test instances of Cordeau Cordeau et al. (1997) PR01-10. The number of customers for Cordeau instances varies between 48 and 288 customers; while all Solomon instances have 100 customers. The author

Aghezzaf and Fahim SpringerPlus (2016)5:1781

in Vansteenwegen (2008) designed OPTW test instances based on Solomon instances by considering the first 50 customers (n = 50). The authors in Montemanni and Gambardella (2009) designed other OPTW test instances based on 27 VRPTW test instances of Solomon namely, C200, R200 and RC200, and on 10 MDVRP test instances of Cordeau PR11-20. We design OPSTW test instances based on OPTW test instance namely PR01 by considering different values of Wmax and Pmax. We set the maximum allowable waiting time to Wmax ∈ [5, 7], in increments of 0.5  %, of the maximum time duration Tmax. While we set the maximum allowable time window violation to Pmax ∈ [1, 5], in increments of 0.5 %, of Tmax. In all OPTW and OPSTW test instances, it is assumed that the maximum time duration Tmax is equal to Tmax = l0 − e0. Parameter tuning

Preliminary experiments on OPTW test instances are conducted to set the values of αi , i ∈ {1, 2, 3, 4}. The following values are tested: αi ∈ [0, 1], i ∈ {1, 2, 3} in increments of 0.1 units and α4 ∈ {1, 2, 3, 4}. The results indicate that given more importance to the metric related to the saving in travel time in the selection criterion has good influence on solution quality. Thus, the following values are selected:

α1 α2 α3 α4

∈ {0.9, 0.7, 0.5} ∈ {0.1, 0.3, 0.5} ∈ {0.1, 0.3, 0.5, 0.7, 0.9} ∈ {2, 3, 4}.

The proposed HILS algorithm has two parameters: a maximum number of iterations allowed Lmax which is ILS stopping condition, and Imax the maximum number of iterations allowed in VNS process. The values of these parameters are determined performing several experiments on a subset of OPSTW test instances that was randomly selected. We test combinations of the following values during the experiments: Lmax ∈ {20, 50, 100}; Imax ∈ {10, 20, 50}. The results of these experiments show that by increasing these parameters better results can be obtained at the expense of extra computational time. In this paper we are looking for a fast algorithm. Thus, we set these parameters to Lmax = 20 and Imax = 10. Computational results on OPTW instances

In this subsection, we compare the results of HILS algorithm with the following state-ofthe-art algorithms: ••  ••  ••  •• 

I3CH: the iterative three-component algorithm of Hu and Lim (2014). IterLS: the iterated local search algorithm of Vansteenwegen et al. (2009). VNS: the variable neighborhood search algorithm of Tricoire et al. (2010). GVNS: the LP-based granular variable neighborhood search algorithm of Labadie et al. (2012). ••  SSA: the slow version of the simulated annealing algorithm of Lin and Yu (2012). ••  FSA: the fast version of the simulated annealing algorithm of Lin and Yu (2012). ••  ILS: the iterated local search algorithm of Gunawan et al. (2015).

Page 11 of 36

Aghezzaf and Fahim SpringerPlus (2016)5:1781

••  GRASP-ELS: the hybrid algorithm of Labadie et al. (2011) that combines greedy randomized adaptive search procedure with evolutionary local search. ••  ABC: the artificial bee colony algorithm of Tunchan (2014). ••  DABC: the discrete artificial bee colony algorithm of Karabulut and Tasgetiren (2013). ••  GA: the genetic algorithm of Karbowska-Chilinska and Zabielski (2014). ••  ACS: the ant colony system algorithm of Montemanni and Gambardella (2009). ••  EACS: the enhanced ant colony system algorithm of Montemanni et al. (2011). ••  VAN: the iterated local search algorithm of Vansteenwegen (2008). The computational results are given in Table  1. Column Instance-n gives the instance over which the algorithms are tested and the associated number of customers. Column BKS presents the latest best known solution as described in (http://centres.smu.edu. sg/larc/orienteering-problem-library). Column I3CH presents the profit obtained by I3CH algorithm (Profit), the percentual gap with the best known profit [Gap (%)] and the computational time required by I3CH algorithm on each run [CPU (s)]. Column ABC presents the average profit, over five runs, obtained by ABC algorithm, the percentual gap with the best known profit and the average computational time of five runs. Column VNS presents the average profit as described in Tricoire et al. (2013), over ten runs, obtained by VNS algorithm, the percentual gap with the best known profit and the average computational time of ten runs. Column ILS presents the profit obtained, over ten runs, by ILS algorithm, the percentual gap with the best known profit and the average computational time of ten runs. Column ACS presents the profit obtained, over five runs, by ACS algorithm, the percentual gap with the best known profit and the average computational time of five runs. Column GVNS presents the profit obtained, over five runs, by GVNS algorithm, the percentual gap with the best known profit and the average computational time of five runs. Column GRASP-ELS presents the profit obtained, over five runs, by GRASP-ELS algorithm, the percentual gap with the best known profit and the average computational time of five runs. Column SSA presents the profit obtained by SSA algorithm, the percentual gap with the best known profit and the computational time of one run. Column IterLS presents the profit obtained by IterLS algorithm, the percentual gap with the best known profit and the computational time of one run. Column FSA presents the profit obtained by FSA algorithm, the percentual gap with the best known profit and the computational time of one run. Column DABC presents the profit obtained, over five runs, by DABC algorithm, the percentual gap with the best known profit and the average computational time of five runs. Column HILS presents the profit obtained, over five runs, by our HILS algorithm, the percentual gap with the best known profit and the average computational time of five runs. Column GA presents the profit obtained, over sixteen runs, by GA algorithm, the percentual gap with the best known profit and the average computational time of sixteen runs. Column Opt presents the optimal profit as described in Tricoire et al. (2010) and the associated computational time. Column VAN presents the profit obtained by VAN algorithm and the associated computational time. Column VNS presents the worst, best and average profit, over ten runs, of VNS algorithm and the average computational time.

Page 12 of 36

247

293

299

308

277

284

297

298

R106-100

R107-100

R108-100

R109-100

R110-100

R111-100

R112-100

0.0

266

RC102-100

266

0.0

219

RC101-100

219

0.6

3.1

0.7

0.0

0.0

0.7

0.7

0.0

0.0

1.7

Avg

289

295

284

277

306

297

293

247

298

0.0

0.0

R105-100

0.0

303

293

286

198

380

0.0

R104-100

380

C109-100

370

0.0

293

370

C108-100

370

0.0

R103-100

370

C107-100

340

0.0

286

340

C106-100

340

0.0

R102-100

340

C105-100

420

0.0

0.0

420

C104-100

400

0.0

0.0

198

400

C103-100

360

320

R101-100

360

C102-100

0.0

320

C101-100

25.5

21.8

28.6

32.0

27.7

33.9

31.1

29.7

27.8

29.4

26.0

27.3

28.8

29.3

20.4

25.2

26.3

24.8

24.7

23.6

23.4

27.1

27.1

28.1

21.8

266

219

297.0

294.0

282.2

277.0

308.0

299.0

289.0

242.6

299.6

292.0

281.6

190.0

380

370

370

340

340

420

398

360

320

Profit

CPU (s)

Profit

Gap (%)

ABC

I3CH

Avg

BKS

Instance-n

Table 1  Results on the OPTW test instances

0.0

0.0

1.0

0.3

1.0

0.6

0.0

0.0

0.0

1.4

1.8

1.1

0.3

1.6

4.2

0.1

0.0

0.0

0.0

0.0

0.0

0.0

0.5

0.0

0.0

Gap (%)

0.6

0.5

3.5

2.4

1.7

8.1

1.6

4.6

7.1

1.6

9.1

1.7

1.1

2.2

0.3

2.6

0.5

0.9

1.3

0.5

0.5

15.2

3.6

0.6

0.4

CPU (s)

266

219

297.9

297.0

284.0

277.0

308

299.0

292.2

247.0

303.0

293.0

285.2

198.0

380

370

370

340

340

418

398

360

320

Profit

VNS

0.0

0.0

0.1

0.0

0.0

0.0

0.0

0.0

0.0

0.3

0.0

0.0

0.0

0.3

0.0

0.1

0.0

0.0

0.0

0.0

0.0

0.5

0.5

0.0

0.0

Gap (%)

54.7

61.2

89.1

97.0

95.4

86.0

77.4

119.1

105.5

89.6

85.4

95.9

90.2

77.9

49.9

98.4

118.7

99.3

88.7

86.7

87.1

124.3

106.2

102.2

72.2

CPU (s)

266

219

298

297

284

277

308

299

293

247

303

293

286

198

380

370

370

340

340

420

400

360

320

Profit

ILS

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

0.0

0.0

0.0

0.0

0.0

Gap (%)

0.4

0.2

1.8

3.3

10.9

1.3

0.2

0.9

0.5

0.2

0.7

1.5

1.4

0.2

0.1

1.0

6.8

0.5

0.1

0.5

0.4

0.4

0.2

0.3

0.2

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 13 of 36

489

595

591

298

463

493

594

PR04-192

PR05-240

PR06-288

PR07-72

PR08-144

PR09-216

PR10-288

960

980

910

930

930

C203-100

C204-100

C205-100

C206-100

C207-100

930

920

900

970

960

0.0

1.1

1.1

1.0

0.0

0.0

0.0

930

C202-100

930

870

C201-100

870

1.1

4.4

0.6

1.9

0.0

0.2

0.2

0.0

0.0

Avg

568

490

454

298

590

594

489

394

2.5

394

12.9

PR03-144

394

305

264

1.1

404

298

RC108-100

274

0.8

PR02-96

277

RC107-100

250

0.0

1.0

252

RC106-100

244

0.0

0.0

308

244

RC105-100

301

266

PR01-48

301

RC104-100

77.4

75.7

70.7

117.4

92.3

87.6

70.1

109.0

222.2

137.8

77.4

26.5

189.9

185.4

109.3

72.9

47.9

20.8

25.6

25.1

26.3

25.0

26.4

27.2

27.1

918

928

900

966

958

930

870

570.4

481.8

462

298

573.6

586.8

486

394

391.2

307

298

276.2

251.4

244

301

266

Profit

1.9

266

RC103-100

ABC CPU (s)

Profit

Gap (%)

I3CH

Avg

BKS

Instance-n

Table 1  continued

1.3

0.2

1.1

1.4

0.2

0.0

0.0

1.5

4.0

2.3

0.2

0.0

2.9

1.4

0.6

0.0

3.2

0.3

0.1

0.0

0.3

0.2

0.0

0.0

0.0

Gap (%)

2.3

6.2

1.7

25.7

14.8

15.3

11.2

78.5

210.3

126.3

15.0

4.3

247.4

91.4

67.4

12.6

4.9

5.8

7.9

11.3

10.4

8.1

1.3

20.3

10.7

CPU (s)

930

927

908

972

960

928

870

575.5

481.5

452.2

297.5

588.2

586.1

488.1

390.5

403.9

308

298

277

252

244

301

266

Profit

VNS

0.0

0.3

0.2

0.8

0.0

0.2

0.0

1.1

3.1

2.3

2.3

0.2

0.5

1.5

0.2

0.9

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

Gap (%)

599.5

616.5

645.4

484

614.3

454.8

507.8

744.0

1534.1

920.8

514.1

131.0

1633.8

1455.2

541.6

387.2

244.6

77.9

65.5

71.5

72.4

67.4

65.2

62.5

69.2

CPU (s)

930

927

908

974

960

930

870





















298

277

252

244

301

266

Profit

ILS

0.0

0.3

0.2

0.6

0.0

0.0

0.0





















0.0

0.0

0.0

0.0

0.0

0.0

0.0

Gap (%)

68.1

111.5

56.2

217.6

137.2

59.0

36.7





















1.0

0.1

0.3

0.3

4.3

0.3

2.0

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 14 of 36

1028

1093

953

1032

1077

1117

959

991

1051

R203-100

R204-100

R205-100

R206-100

R207-100

R208-100

R209-100

R210-100

R211-100

1003

1140

859

899

983

1057

RC203-100

RC204-100

RC205-100

RC206-100

RC207-100

RC208-100

442

PR12-96

2.0

0.0

433

2.8

353

353

PR11-48

5.1

2.6

4.0

1.4

4.1

3.7

Avg

1003

957

863

847

1093

966

1.5

938

RC202-100

924

0.0

795

RC201-100

795

1.4

3.6

0.9

1.1

2.2

2.5

1.1

0.7

1.8

0.8

Avg

1013

982

948

1092

1050

1021

946

1073

1020

0.0

930

R202-100

930

1.0

789

797

0.0

R201-100

950

59.8

30.8

119.3

123.0

122.0

98.4

99.2

167.5

134.3

129.3

80.9

176.2

167.4

176.9

136.5

230.0

192.8

169.3

129.3

236.9

221.4

175.6

101.8

84.4

84.0

432

350.6

1032.8

977.0

890.2

852.3

1109.2

962.4

926.8

784

1019

965.4

934.6

1084.6

1060.8

1018

953

1070.4

1009

895.8

787

950

Profit

0.4

950

C208-100

ABC CPU (s)

Profit

Gap (%)

I3CH

Avg

BKS

Instance-n

Table 1  continued

2.3

0.7

1.8

2.3

0.6

1.0

0.8

2.7

4.0

1.2

1.4

2.1

3.0

2.6

2.5

2.9

1.5

1.4

0.0

2.1

1.8

3.7

1.3

0.5

0.0

Gap (%)

23.5

1.7

24.3

26.3

15.8

24.9

27.2

23.4

24.8

27.3

24.9

26.6

15.0

29.3

26.3

26.7

20.9

21.3

32.9

34.0

25.6

40.6

20.3

11.1

11.8

CPU (s)

442

328

1043.5

960.8

878.4

845.9

1120.8

988.2

925.6

795

1024.2

975.9

947.8

1101.5

1061.5

1014

952.6

1076.3

1007.9

905.6

796.7

949

Profit

VNS

0.0

7.1

1.5

1.3

2.3

2.3

1.5

1.7

1.5

1.3

0.0

1.4

2.5

1.5

1.2

1.4

1.4

1.7

0.0

1.5

2.0

2.6

0.0

0.2

0.1

Gap (%)

309.6

131.4

869.4

995.5

893.5

838.7

726.6

970.3

938.4

951.5

640.8

1065.8

1120.7

1065.6

925.9

1160.8

1065.3

1168.1

745.1

1253.5

1139.9

1057.5

1021.7

560.2

558.9

CPU (s)





1040.7

952.1

894.1

854.7

1131.3

989.8

929

795

1040.4

970.8

949.7

1107.6

1059.5

1012.4

948.4

1082.8

1011.3

910.3

788.7

950

Profit

ILS





1.1

1.5

3.1

0.5

0.5

0.8

1.3

1.0

0.0

1.3

1.0

2.0

1.0

0.8

1.6

1.9

0.5

0.9

1.6

2.1

1.0

0.1

0.0

Gap (%)





120.5

85.6

129.9

152.0

100.5

165.0

111.5

156.2

63.5

162.1

145.7

171.8

145.8

165.6

174.0

126.5

169.9

171.0

213.5

165.6

133.7

90.0

33.3

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 15 of 36

467

567

708

674

362

539

562

667

PR13-144

PR14-192

PR15-240

PR16-288

PR17-72

PR18-144

PR19-216

PR20-288

380

C109-100

292.6

286

198

380

370 0.0

0.0

0.0

0.1

0.0

293

370

C108-100

370

0.0

R103-100

370

C107-100

340

0.0

286

340

C106-100

340

0.0

R102-100

340

C105-100

420

0.0

0.0

420

C104-100

400

0.0

0.0

198

400

C103-100

360

320

R101-100

360

C102-100

0.0

320

C101-100

640.6

11.1

0.1

6.4

0.8

0.8

2.1

1.0

0.9

33.5

16.9

0.7

0.5

CPU (s)

286

274.8

197

380

354

358

340

340

410

396

360

320

Profit

Gap (%)

604.8

541

535

359

613.4

666.4

560.1

458.1

Profit

130.2

202.7

164.6

99.0

34.7

228.6

248.2

144.4

89.5

GVNS

4.3

11.8

4.3

7.8

0.6

8.2

0.1

8.1

ACS

588

538

497

360

619

707

521

0.2

Profit

466

ABC CPU (s)

Profit

Gap (%)

I3CH

Avg

BKS

Instance-n

Avg

BKS

Instance-n

Table 1  continued

2.4

4.1

0.5

1.3

0.0

4.5

3.4

0.0

0.0

2.4

1.0

0.0

0.0

Gap (%)

3.6

9.3

3.7

0.7

0.8

9.0

5.9

1.2

1.9

Gap (%)

33.8

13.4

0.2

166.5

30.6

1.1

0.7

6.5

3.3

1005.9

382.4

67.4

0.2

CPU (s)

103.7

346.9

70.7

22.1

6.6

205.1

274.6

76.6

8.9

CPU (s)

290.4

286

198

380

370

370

340

340

420

400

360

320

Profit

GRASP-ELS

647.8

551.6

519.4

358.3

637

663.2

550.1

453.7

Profit

VNS

0.9

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

Gap (%)

3.4

2.9

1.9

3.6

1.0

5.5

6.3

3.0

2.8

Gap (%)

4.8

2.3

0.9

22.6

25.5

6.6

3.7

3.8

2.3

105.2

38.5

16.5

1.2

CPU (s)

1045.9

2441.9

1242.2

760.9

140.5

1854.1

1900.9

1163.8

514.1

CPU (s)

293

286

198

380

370

370

340

340

420

400

360

320

Profit

SSA

















Profit

ILS

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

Gap (%)

















Gap (%)

20.3

21

19.7

21.1

20.5

20.5

20.5

20.3

20.6

21.9

24.2

20.7

20.4

CPU (s)

















CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 16 of 36

244

252

277

298

RC106-100

RC107-100

RC108-100

394

489

595

591

PR03-144

PR04-192

PR05-240

PR06-288

564.6

576.8

482.6

394

4.5

3.1

1.3

0.0

0.0

404

PR02-96

403.8

0.0

308

PR01-48

308

0.0

0.0

0.0

0.0

0.0

0.0

Avg

298

277

252

244

301

0.0

0.0

RC105-100

0.2

301

266

266

219

297.4

0.1

RC104-100

298

R112-100

296.6

0.3

266

297

R111-100

283.2

0.0

0.7

RC103-100

284

R110-100

277

306

266

277

R109-100

1.5

RC102-100

308

R108-100

294.6

0.0

0.0

299

R107-100

293

0.0

0.0

219

293

R106-100

247

303

RC101-100

247

R105-100

0.2

303

R104-100

2199.8

2075.7

1404.7

2024.7

1147.8

256.2

143.2

207.5

502.1

308.6

9.7

29.4

57.2

30.9

0.2

383.4

947.1

484.4

617.6

28.0

696.1

922.6

86.3

3.0

164.0

584.2

578

475.4

388

403.6

307.2

288.8

261.2

252

227

301

253.2

246

219

289.6

290.6

279.2

276.4

301.4

287.2

280.4

230.6

298.6

Profit

CPU (s)

Profit

Gap (%)

GVNS

ACS

Avg

BKS

Instance-n

Table 1  continued

1.2

2.9

2.8

1.5

0.1

0.3

3.7

3.2

6.0

0.0

7.5

0.0

5.1

8.1

0.0

2.8

2.9

2.2

1.7

0.2

2.2

4.1

4.5

7.1

1.5

Gap (%)

29.0

20.5

14.7

4.1

3.7

1.2

9.8

10.4

15.7

2.2

2.7

16.2

23.3

5.8

2.1

29.4

31.9

54.2

16.8

10.4

50.1

50.3

19.1

0.3

72.7

CPU (s)

583.4

581

474.6

394

402.6

308

298

277

252

244

300.2

265.2

259

219

297.2

297

283.6

277

306.8

296.4

293

247

302.8

Profit

GRASP-ELS

1.3

2.4

2.9

0.0

0.3

0.0

0.4

0.0

0.0

0.0

0.0

0.3

0.3

2.7

0.0

0.2

0.3

0.0

0.1

0.0

0.4

0.9

0.0

0.0

0.1

Gap (%)

8.6

9.1

5.3

3.1

3.0

1.2

2.0

2.9

1.9

1.2

1.4

3.1

2.7

2.1

0.6

3.5

4.6

4.6

2.8

2.7

5.0

4.8

3.3

1.2

5.1

CPU (s)

575

589

489

394

404

305

298

277

252

244

301

266

266

219

298

297

284

277

306

297

293

247

303

Profit

SSA

2.7

1.0

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

0.1

0.0

0.0

0.0

0.0

0.7

0.7

0.0

0.0

0.0

Gap (%)

253.4

281.7

106.7

59.9

29.1

8.3

22.2

26

22

21

20.3

27.5

20.7

20.2

19.8

23.3

22.3

27.9

20.8

20.4

40.6

21.5

22.1

20.3

23.2

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 17 of 36

960

980

910

930

930

950

C204-100

C205-100

C206-100

C207-100

C208-100

1028

1093

953

1032

1077

1117

959

991

1051

R203-100

R204-100

R205-100

R206-100

R207-100

R208-100

R209-100

R210-100

R211-100

1006.8

956.4

923.4

1057.2

1026.6

983.0

939.2

1046.4

989.9

4.2

3.5

3.7

5.4

4.7

4.7

1.4

4.3

3.7

3.3

930

R202-100

899.6

0.2

797

R201-100

795.8

0.6

1.0

1.1

1.1

0.4

1.0

0.0

Avg

940.2

920

920

906

970

960

0.0

C203-100

930

930

0.9

C202-100

870

588.4

2.3

0.0

594

PR10-288

481.8

0.1

0.0

1.2

493

PR09-216

462.6

298

870

463

PR08-144

C201-100

298

PR07-72

1288.1

979.1

1453.2

1260.9

1090.3

1671.4

1125.4

1549.1

2641.7

1724.4

2339.9

342.6

89.0

48.5

87.9

28.2

1617.5

361.0

319.0

189.8

1626.7

2343.5

2318.2

2476.0

20.1

1025.6

961.4

926

1083.6

1022

966.6

905.8

1073.8

992.2

881.4

775.6

942

928

922

898

966

956

916

850

564.4

482

463

297

Profit

CPU (s)

Profit

Gap (%)

GVNS

ACS

Avg

BKS

Instance-n

Table 1  continued

2.4

3.0

3.4

3.0

5.1

6.3

5.0

1.8

3.5

5.2

2.7

1.1

0.8

0.2

0.9

1.3

1.4

0.4

1.5

2.3

1.6

5.0

2.2

0.0

0.3

Gap (%)

22.5

30.1

37.3

52.9

46.6

34.9

15.3

77.6

34.7

13.4

6.7

192.4

74.2

36.3

34.4

12.5

974

329

78.7

0.1

12.4

32.7

12.3

3.8

1.7

CPU (s)

1028.2

969

937.4

1100.2

1049.2

1023.6

927.2

1071.8

1001.4

909.8

788.2

942

930

926

906

970

960

910

870

574.2

477.8

462.8

294.4

Profit

GRASP-ELS

2.2

2.2

2.3

1.5

2.6

0.8

2.7

1.9

2.6

2.2

1.1

0.6

0.8

0.0

0.4

0.4

1.0

0.0

2.2

0.0

1.5

3.3

3.1

0.0

1.2

Gap (%)

11.3

10.8

10.3

15.9

13.5

10.7

7.8

15.2

12.6

9.6

5.3

32.2

17.7

17.3

13.4

10.6

130.3

45.2

20.4

2.5

5.0

8.1

7.0

3.3

1.6

CPU (s)

1041

973

945

1079

1069

1020

946

1058

997

914

794

950

930

930

910

970

960

930

870

578

482

462

298

Profit

SSA

1.0

1.8

1.5

3.4

0.7

1.2

0.7

3.2

3.0

1.7

0.4

0.1

0.0

0.0

0.0

0.0

1.0

0.0

0.0

0.0

1.0

2.7

2.2

0.2

0.0

Gap (%)

40.5

53.6

61.6

35.8

52.5

40.9

37.8

39.7

44.2

46.4

51.1

37.5

31.2

29.1

29

46.9

42.3

59.6

33.5

28.3

112.2

189.7

102.3

76.0

15.0

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 18 of 36

938

1003

1140

859

899

983

1057

RC202-100

RC203-100

RC204-100

RC205-100

RC206-100

RC207-100

RC208-100

567

708

674

362

539

562

667

PR14-192

PR15-240

PR16-288

PR17-72

PR18-144

PR19-216

PR20-288

Avg

467

PR13-144

552.4

475.0

488.8

358.0

517.8

524.8

494.0

441.0

11.9

17.2

15.5

9.3

1.1

23.2

25.9

12.9

5.6

1.3

442

PR12-96

436.4

7.1

353

PR11-48

327.8

2.2

4.1

2.3

1.7

1.2

4.7

2.6

Avg

1013.2

960.8

884.0

848.4

1086.2

976.6

0.7

0.0

931.6

3.5

795

795

887.7

24.6

30.0

1350.0

1330.9

25.7

18.2

23.1

2312.7

2017.6

1743.7

869.4

995.5

893.5

838.7

726.6

970.3

938.4

951.5

640.8

1556.7

655.0

507.8

530.8

354.6

643.4

656.6

540.6

452.4

435

329

1000.2

911.4

866.6

843.8

1101.8

954.4

890.6

784

Profit

CPU (s)

Profit

Gap (%)

GVNS

ACS

RC201-100

BKS

Avg

Instance-n

Table 1  continued

4.3

1.8

9.6

1.5

2.0

4.5

7.3

4.7

3.1

1.6

6.8

4.1

5.4

7.3

3.6

1.8

3.4

4.8

5.1

1.4

3.8

Gap (%)

24.2

57.0

22.1

10.4

5.4

60.9

29.4

32.4

16.2

6.5

1.9

16.0

24.6

14.7

11.6

7

31.9

22.5

12.1

3.7

33.8

CPU (s)

656.6

537.8

517.6

356.2

652.6

665.0

541.2

456.6

442.0

329.2

1007.0

965.4

886.8

844.6

1096.0

978.2

922.8

786.4

Profit

GRASP-ELS

3.4

1.6

4.3

4.0

1.6

3.2

6.1

4.6

2.2

0.0

6.7

2.3

4.7

1.8

1.4

1.7

3.9

2.5

1.6

1.1

2.0

Gap (%)

7.9

13.9

8.7

5.0

2.4

16.2

13.1

8.8

4.9

4.0

2.0

8.2

10.7

7.5

6.1

6.8

11.9

9.6

8.9

4.2

11.2

CPU (s)

626.0

531.0

539.0

362.0

616.0

666.0

540.0

452.0

430.0

351.0

1041

977

885

854

1140

967

930

795

Profit

SSA

3.7

6.1

5.5

0.0

0.0

8.6

5.9

4.8

3.2

2.7

0.6

1.1

1.5

0.6

1.6

0.6

0.0

3.6

0.9

0.0

1.7

Gap (%)

162.4

475.3

152.9

61.8

37.6

558.0

118.5

134.3

49.0

26.3

10.3

50.3

51.2

68.4

60.2

52.6

46.5

32.4

46.3

44.4

45.8

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 19 of 36

293

299

308

277

284

297

298

R106-100

R107-100

R108-100

R109-100

R110-100

R111-100

R112-100 0.0

219

RC101-100

1.0

0.7

1.1

0.4

3.7

3.8

0.0

0.0

2.0

295

295

281

276

297

288

293

247

2.0

Avg

219

247

297

2.4

0.0

R105-100

0.0

303

286

286

182

380

0.0

R104-100

380

C109-100

370

2.8

0.0

293

370

C108-100

360

340

R103-100

370

C107-100

0.0

286

340

C106-100

340

5.0

R102-100

340

C105-100

400

2.6

8.8

420

C104-100

390

0.0

0.0

198

400

C103-100

360

320

R101-100

360

C102-100

1.2

320

C101-100

0.2

0.2

0.2

0.2

0.3

0.2

0.2

0.2

0.2

0.1

0.2

0.2

0.2

0.1

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.5

0.3

0.4

216

284

286

282

264

294

277

270

247

294

293

282

198

380

370

370

340

330

410

390

360

320

Profit

CPU (s)

FSA Gap (%)

IterLS

Profit

Avg

BKS

Instance-n

Table 1  continued

1.4

3.2

4.7

3.7

0.7

4.7

4.5

7.4

7.8

0.0

3.0

0.0

1.4

0.0

0.9

0.0

0.0

0.0

0.0

2.9

2.4

2.5

0.0

0.0

Gap (%)

0.2

0.2

0.2

0.2

0.3

0.2

0.2

0.2

0.2

0.1

0.2

0.2

0.2

0.1

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.5

0.3

0.4

CPU (s)

206

290.8

295

281.2

233.4

306

297

293

243.4

303

293

286

182.2

380

370

350

340

322

420

390

360

312

Profit

DABC

5.9

2.6

2.4

0.7

1.0

15.7

0.6

0.7

0.0

1.5

0.0

0.0

0.0

8.0

1.7

0.0

0.0

5.4

0.0

5.3

0.0

2.5

0.0

2.5

Gap (%)

2.2

25.3

50.8

85.4

1.0

0.6

31

1.4

0.8

62.2

15

49.4

5.4

0.6

17.3

22.0

18.8

1.2

2.0

1.2

10.4

1.0

98.6

0.4

CPU (s)

219

295

294

281

276

308

299

293

247

299

287

286

198

380

370

370

340

340

420

390

360

320

Profit

HILS

0.0

0.6

1.0

1.0

1.1

0.4

0.0

0.0

0.0

0.0

1.3

2.0

0.0

0.0

0.3

0.0

0.0

0.0

0.0

0.0

0.0

2.5

0.0

0.0

Gap (%)

0.2

0.6

0.7

0.7

0.4

0.4

0.7

0.7

0.6

0.3

0.7

0.7

0.6

0.2

0.4

0.4

0.4

0.4

0.4

0.4

0.7

0.4

0.4

0.4

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 20 of 36

489

595

591

298

463

493

594

PR05-240

PR06-288

PR07-72

PR08-144

PR09-216

PR10-288

960

980

910

930

C203-100

C204-100

C205-100

C206-100

910

900

950

940

2.2

1.1

3.1

2.1

2.2

930

C202-100

910

3.4

870

C201-100

840

4.7

9.3

6.5

0.0

2.3

9.0

3.2

8.6

2.5

Avg

539

461

463

291

538

576

447

384

4.7

PR04-192

3.5

394

385

304

288

1.1

PR03-144

298

RC108-100

274

5.4

404

277

RC107-100

239

10.4

PR02-96

252

RC106-100

221

1.3

1.3

244

RC105-100

297

0.4

2.7

308

301

RC104-100

265

259

PR01-48

266

RC103-100

3.1

266

RC102-100

1.6

1.2

1.6

1.7

2.8

1.1

1.8

3.6

1.4

1.0

0.4

2.5

4.6

1.9

1.0

0.6

0.5

0.2

0.2

0.2

0.2

0.2

0.3

0.3

0.2

920

900

950

940

930

870

539

461

438

289

557

527

470

381

392

304

263

244

240

219

263

265

249

Profit

CPU (s)

FSA Gap (%)

IterLS

Profit

Avg

BKS

Instance-n

Table 1  continued

1.1

1.1

3.1

2.1

0.0

0.0

5.3

9.3

6.5

5.4

3.0

5.8

11.4

3.9

3.3

3.0

1.3

7.4

11.7

11.9

4.8

10.2

12.6

0.4

6.4

Gap (%)

1.6

1.2

1.6

1.7

2.8

1.1

1.8

3.6

1.4

1.0

0.4

2.5

4.6

1.9

1.0

0.6

0.5

0.2

0.2

0.2

0.2

0.2

0.3

0.3

0.2

CPU (s)

928

906

970

960

930

840





















284

262.4

225

228

288.4

255.8

259

Profit

DABC

0.2

0.4

1.0

0.0

0.0

3.4





















5.5

4.7

5.3

10.7

6.6

4.2

3.8

2.6

Gap (%)

142

23.6

117.6

153.4

127.4

23.8





















28.6

82.6

45

0.6

4.4

86.8

6.4

1.0

CPU (s)

920

900

970

960

910

870

526

447

455

285

496

523

452

378

397

308

278

276

244

244

275

248

259

Profit

HILS

1.1

1.1

1.0

0.0

2.2

0.0

6.8

11.4

9.3

1.7

10.7

16.1

12.4

7.6

4.1

1.7

0.0

3.5

6.7

0.4

3.2

0.0

8.6

6.8

2.6

Gap (%)

5.9

5.3

9.2

6.2

5.7

3.9

2.7

4.9

2.2

2.1

0.4

5.8

5.5

2.1

1.4

1.2

1.1

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.3

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 21 of 36

1028

1093

953

1032

1077

1117

959

991

1051

R203-100

R204-100

R205-100

R206-100

R207-100

R208-100

R209-100

R210-100

R211-100

1140

859

899

983

1057

RC204-100

RC205-100

RC206-100

RC207-100

RC208-100 6.5

330

PR11-48

1.9

5.8

4.3

2.2

2.0

4.3

3.6

1037

926

860

840

1117

960

Avg

353

1003

RC203-100

6.0

938

RC202-100

882

1.9

795

RC201-100

780

3.3

2.7

3.3

3.4

4.3

3.6

3.5

2.3

1.8

4.7

Avg

1023

958

926

1069

1038

996

931

1073

980

5.4

930

R202-100

880

1.1

788

2.1

2.2

797

930

910

R201-100

950

C208-100 2.3

930

C207-100

0.3

1.6

2.3

1.3

1.1

1.0

2.3

2.7

1.3

1.0

1.7

1.6

1.9

2.4

1.6

2.0

1.5

1.4

1.7

1.6

1.4

1.2

1.7

1.6

2.1

339

1015

899

866

800

1061

956

866

781

1023

975

914

1066

1033

950

936

1059

988

878

772

940

930

Profit

CPU (s)

FSA Gap (%)

IterLS

Profit

Avg

BKS

Instance-n

Table 1  continued

4.0

5.5

4.0

8.5

3.7

6.9

6.9

4.7

7.7

1.8

3.9

2.7

1.6

4.7

4.6

4.1

7.9

1.8

3.1

3.9

5.6

3.1

1.1

1.1

0.0

Gap (%)

0.3

1.6

2.3

1.3

1.1

1.0

2.3

2.7

1.3

1.0

1.7

1.6

1.9

2.4

1.6

2.0

1.5

1.4

1.7

1.6

1.4

1.2

1.7

1.6

2.1

CPU (s)



1041.0

937.6

884.8

841.2

1130.8

983.4

924.4

792

1037.2

972

948.4

1114

1070.6

1024.8

951.4

1089.4

1019.6

912.8

784

950

930

Profit

DABC



1.8

1.5

4.6

1.6

2.1

0.8

2.0

1.4

0.4

1.0

1.3

1.9

1.1

0.3

0.6

0.7

0.2

0.3

0.8

1.8

1.6

0.6

0.0

0.0

Gap (%)



107.4

34.0

117.2

81.4

119.2

168.8

138.0

135.2

65.6

135.6

116.8

170.4

164.4

130.2

142.2

187.8

129.0

77.4

218.6

105.4

49.4

107.8

132.2

142.4

CPU (s)

351

1008

923

851

820

1068

947

905

771

996

929

892

1061

1015

994

894

1037

970

890

781

950

920

Profit

HILS

0.6

4.9

4.6

6.1

5.3

4.5

6.3

5.6

3.5

3.0

5.1

5.2

6.3

7.0

5.0

5.8

3.7

6.2

5.1

5.6

4.3

2.0

0.8

0.0

1.1

Gap (%)

0.7

9.0

13.3

9.2

6.1

6.0

19.4

10.3

6.2

1.8

19.2

21.5

16.3

14.3

33.7

23.5

18.4

11.2

29.6

22.4

14.3

6.1

6.0

6.1

6.1

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 22 of 36

442

467

567

708

674

362

539

562

667

PR12-96

PR13-144

PR14-192

PR15-240

PR16-288

PR17-72

PR18-144

PR19-216

PR20-288

290

286

189

380

0.0

0.0

1.0

0.0

293

380

C109-100

370

2.2

0.0

0.0

4.0

2.8

0.0

0.9

R103-100

370

C108-100

362



















CPU (s)

Gap (%)

286

370

C107-100

340



















R102-100

340

C106-100

340



















4.5

340

C105-100

403

389

360

317

Profit

GA

2.0

2.5

2.7

0.8

0.2

4.1

5.3

1.1

1.9

0.9

Gap (%)

198

420

C104-100

10.2

14.8

10.9

11.5

7.5

16.6

13.4

12.0

9.2

2.3

Profit

R101-100

400

C103-100

568

501

477

335

562

613

499

424

432

CPU (s)

1.1

360

C102-100

2.0

2.5

2.7

0.8

0.2

4.1

5.3

1.1

1.9

0.9

Gap (%)

DABC

Avg

320

C101-100

9.6

14.5

11.2

11.1

4.4

17.1

9.9

15.0

3.6

2.5

BKS

570

499

479

346

559

638

482

450

431

Profit

CPU (s)

Profit

Gap (%)

FSA

IterLS

Instance-n

Avg

BKS

Instance-n

Table 1  continued

588

479

459

363

582

616

479

436

427

Profit

HILS

9.4

11.8

14.8

14.8

−0.3

13.6

13.0

15.5

6.6

3.4

Gap (%)

0.3

0.3

0.2

0.3

0.3

0.3

0.3

0.3

0.3

0.4

0.4

0.3

0.3

CPU (s)

6.3

10.2

6.2

4.1

1.0

13.3

16.4

6.0

2.7

2.1

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 23 of 36

244 252 277 298

RC106-100

RC107-100

RC108-100

960 980 910 930 930 950

C203-100

C204-100

C205-100

C206-100

C207-100

C208-100

931

910

905

893

952

931

2.0

2.2

2.7

1.9

2.9

3.0

3.1

930

C202-100

901

2.5

870

C201-100

848

2.4

3.0

2.9

2.8

3.3

2.3

Avg

289

269

245

236

294

1.5

1.9

RC105-100

1.7

301

262

261

216

293

1.3

RC104-100

298

R112-100

293

2.5

266

297

R111-100

277

2.5

RC103-100

284

R110-100

270

2.6

266

277

R109-100

300

2.3

RC102-100

308

R108-100

292

0.3

1.4

299

R107-100

292

1.2

2.0

219

293

R106-100

244

297

RC101-100

247

R105-100

Gap (%)

1.8

303

R104-100

Profit

GA

Avg

BKS

Instance-n

Table 1  continued

1.0

1.0

1.0

1.0

1.5

1.1

1.0

0.8

0.3

0.3

0.3

0.3

0.3

0.4

0.3

0.3

0.2

0.3

0.4

0.4

0.3

0.3

0.3

0.4

0.3

0.2

0.3

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 24 of 36

930 1028 1093 953 1032 1077 1117 959 991 1051

R202-100

R203-100

R204-100

R205-100

R206-100

R207-100

R208-100

R209-100

R210-100

R211-100

1140 859 899 983 1057

RC204-100

RC205-100

RC206-100

RC207-100

RC208-100

Avg

1003

RC203-100

976

896

850

808

1044

866

7.6

7.7

8.9

5.5

5.9

8.4

13.7

7.7

938

RC202-100

866

3.4

795

RC201-100

768

7.1

8.0

7.7

6.5

6.7

8.4

7.6

8.1

6.9

7.2

Avg

967

915

897

1042

986

954

876

1018

954

6.8

4.6

867

2.5 760

Gap (%)

797

Profit

GA

R201-100

BKS

Avg

Instance-n

Table 1  continued

1.5

1.4

1.7

1.3

1.3

2.0

1.3

1.3

1.3

2.0

2.4

1.8

1.7

2.4

2.1

2.0

1.6

2.5

2.2

1.8

1.2

1.0

CPU (s)

Aghezzaf and Fahim SpringerPlus (2016)5:1781 Page 25 of 36

300

320

340

300

280

310

320

340

C103-50

C104-50

C105-50

C106-50

C107-50

C108-50

C109-50

159

208

220

227

192

208

223

226

R105-50

R106-50

R107-50

R108-50

R109-50

R110-50

R111-50

R112-50 0.0 180

0.2

0.2

0.2

0.2

0.2

0.2

0.3

0.2

0.1

0.3

0.2

0.1

0.0

226

223

208

192

223

220

203

159

227

210

Adjusted CPU (s)

50.9

3.1

1.8

1.6

0.2

410.6

9.8

0.6

0.0

152.7

30.0

RC101-50

180

227

R104-50

Avg CPU (s)

214

0.2

R103-50

195

198

R102-50

0.7

0.1 0.1

126

0.3

0.2

0.3

0.2

0.2

0.3

0.3

0.3

0.3

0.3

126

0.0

340

320

310

280

300

340

320

300

270

Adjusted CPU (s)

11.8

0.9

0.2

0.0

0.0

0.0

81.4

23.0

1.1

0.0

CPU (s)

R101-50

Avg CPU (s)

270

C102-50

Profit

Profit

CPU (s)

VAN

Opt

C101-50

Instance-n

Table 1  continued

180

226

223

208

192

227

219

208

159

227

214

198

126

340

320

310

280

300

330

320

300

270

Worst

VNS

180

226

223

208

192

227

220

208

159

227

214

198

126

340

320

310

280

300

340

320

300

270

Best

180

226

223

208

192

227

219.9

208

159

227

214

198

126

340

320

310

280

300

340

320

300

270

Avg

30.8

Iterated local search algorithm for solving the orienteering problem with soft time windows.

In this paper we study the orienteering problem with time windows (OPTW) and the impact of relaxing the time windows on the profit collected by the ve...
1MB Sizes 0 Downloads 16 Views