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 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.

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. (2011Labadie et al. ( , 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.
• 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 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 p i (p 0 = 0), a service time S i (S 0 = 0) and a time window [e i , l i ] which is assumed to be hard. Each arc (i, j) ∈ E has an associated cost t ij which represents the time required to travel from vertex i to vertex j. The cost t ij is defined as the Euclidean distance between the points corresponding to i and j. The arrival time to a customer i is denoted a i ; the beginning of service time is denoted b i . The objective is to design a route R that maximizes the total collected profit subject to the following: • The route R cannot start before e 0 and cannot end after l 0 .
• The service to a customer i cannot start before e i and if the vehicle arrives too early it can wait for a certain period of time w i = max(e i − a i , 0) and serves that customer. • Every customer is visited at most once.
• The total travel time of R is limited by a time limit T max .
In order to formulate the model we define the following decision variables: x ij binary variable equal to 1 if the vehicle travels directly from vertex i to vertex j, and 0 otherwise. y i binary variable equal to 1 if vertex i is visited, and 0 otherwise. b i 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: ( 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 [e i , l i + P max ] = [e i ,l i ], where P max is an upper bound on the maximum allowable time window violation. An appropriate penalty P penalty i is then paid if the service starts late that is a i ∈]l i ,l i ]. The penalty function can be defined as follows: 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: Since in our formulation we take into account the fact that w i ≤ W max for each routed customer i, the following constraint is added to the model.
Regarding the time windows constraints they change as follows: In the next subsection, we will describe the approach that we propose to deal with the OPSTW.

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 Ś . If Ś satisfies an acceptance criterion, it replaces the starting solution and the next perturbation phase is performed from that solution. Otherwise, Ś is discarded and the search returns to the previous starting solution. In the proposed ILS algorithm, a variable neighborhood search (VNS) is applied to Ś even if it is better than S or not. The objective is intensifying the search around Ś , 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 L max . 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 = (i 0 , i 1 , i 2 , . . . , i m−1 , i m ), i 0 = i m = 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 (i p−1 , i p ) p ∈ {1, . . . , m} in R for which, if u is inserted between the adjacent vertices i p−1 and i p then: The waiting time at customer u given by Eq. (15) satisfies the following: The time duration constraint on the route is still satisfied, that is: 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: where: The metric b new i p corresponds to the beginning of service at customer i p (which is the arrival time at the depot if p = m) given that customer u is inserted between i p−1 and i p . This metric is computed as follows: It is clear that PF i p ≥ 0 since the matrix (t ij ) (i,j)∈E satisfies the triangle inequality. This metric is computed for the rest of the subsequent vertices as follows:

Insertion heuristics
Each insertion heuristic H l , 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: The first insertion heuristic H 1 computes this position using the following criterion: The second insertion heuristic H 2 computes this position using the following criterion: While the third insertion heuristic H 3 computes this position using the following criterion: The metric θ u corresponds to the time difference between the completion of service at customer i p−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: The weights α 1 , α 2 and α 3 define the relative contribution of each individual metric in the computing of the best feasible insertion place. The parameter α 1 takes into account the saving in travel time by inserting u between i p−1 and i p . 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.
where ∇ is the set of unrouted customers having at least one feasible place on the current partial route. The criterion C 2 (i(u), u, j(u)) is expressed as follows: The parameter α 4 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 α i , 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 α i , i ∈ {1, 2, 3, 4} are positive weights that satisfy: α i ≥ 0, i ∈ {1, 2, 3, 4} and α 1 + α 2 = 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 W max . This is solved using local search procedure that will be described in the next subsection.

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 Ŝ . The local-search procedure performs, around Ŝ , 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 Ŝ .
• a routed customer in S cannot be inserted in Ŝ .
• the position of a customer in S can be changed to another position in Ŝ .
• the customers of Ŝ 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 Ś is randomly generated from the neighborhood of S. A local search is then applied with Ś as the initial solution, the obtained solution is denoted Ŝ . If Ŝ is better than S, the search moves to Ŝ 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 ϕ S , ϑ S ∈ N. To do that, let us denote with S = (0, c 1 , c 2 , . . . , c m , 0) c h ∈ C, h ∈ {1, . . . , m} a feasible solution. These metrics are defined as follows: The first neighborhood of S denoted N 1 (S) is defined as the set of all solutions obtained by removing two customers, one from the set (c 1 , c 2 , . . . , c ϕ S ) and the other from the set (c ϑ S , c ϑ S +1 , . . . , c m ). The second neighborhood N 2 (S) is defined as the set of all solutions obtained by removing a set of customers (c k : 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 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 W max and P max . We set the maximum allowable waiting time to W max ∈ [5, 7], in increments of 0.5 %, of the maximum time duration T max . While we set the maximum allowable time window violation to P max ∈ [1, 5], in increments of 0.5 %, of T max . In all OPTW and OPSTW test instances, it is assumed that the maximum time duration T max is equal to T max = l 0 − e 0 .

Parameter tuning
Preliminary experiments on OPTW test instances are conducted to set the values of The following values are tested: 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: The proposed HILS algorithm has two parameters: a maximum number of iterations allowed L max which is ILS stopping condition, and I max 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: L max ∈ {20, 50, 100}; I max ∈ {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 L max = 20 and I max = 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). • 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).
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.

Table 1 continued
The new best solutions obtained by HILS algorithm are presented as italics numbers Column HILS presents the worst, best and average profit, over five runs, of HILS algorithm and the average computational time.
To ensure fair comparisons, the results of each algorithm are compared with the best known solutions with the computational times adjusted to the speed of the computers used to achieve these results. We summarize the experimental environment of each algorithm and compare their CPU speed in Table 2. As all algorithms are single threaded, we compare their CPU speed using the Super pi benchmark (http://www.superpi.net/). In Table 2, the Super pi column reports the number of seconds it takes each processor to compute the first one million digits of π. The processor used for IterLS algorithm is approximately two times faster than that used for GRASP-ELS and GVNS algorithms; the processor used for VAN algorithm is slower than that used for GRASP-ELS and GVNS algorithms. While the processor used for ACS and EACS algorithms is comparable to that used for GRASP-ELS and GVNS algorithms. This statement is based on the comparison of various computer systems solving standard linear equation problems presented in Dongarra (2014). The performance is evaluated on a benchmark problem of a dense system of linear equations given by a matrix of order 100 and 1000. The values for the processor used for GRASP-ELS and GVNS algorithms are 1571 and 3650 Mflop/s respectively; the values for the processor used for ACS and EACS algorithms are 1470 and 3654 Mflop/s respectively. The values for the processor used for IterLS algorithm are 2426 and 7519 Mflop/s respectively while the values for the processor used for VAN algorithm are 1317 and 2444 Mflop/s respectively. The values for the processor used for VNS which is an Intel Pentium 4 with 2.4 GHz are not available. However, the values for an Intel Pentium 4 with 2.5 GHz are available, this processor has achieved 1190 and 2355 Mflop/s respectively. Thus, we assume that the processor used for VNS algorithm is slower than that used for VAN algorithm. The values for the processor used for GA algorithm are not available. As only limited information was available on the processor used by GA, we cannot estimate its speed by considering only the clock-rate. A review of all processors i7 with 2.93 GHz shows that the Super pi ranges from 13.8 to 11.7 s. We assume that the processor used for DABC is comparable to that used for I3CH algorithm. We estimate the single-thread performance of each processor by supposing the performance of the machine of Gunawan et al. (2015) to be 1. Table 3 summarizes the results obtained by the algorithms. It compares the average performance of HILS with that of the stat-of-the-art algorithms. Column Gap (%) reports the average percentual gap with the best known profit. Column CPU (s) reports the average computational time in seconds. The computational times of the algorithms are adjusted according to computers' speed as presented in Table 2. The authors in Montemanni et al. (2011) did not report detailed results for their EACS algorithm, they have just reported average gaps with the former best known solutions. However, the authors in Gunawan et al. (2015) have recently reported the average gaps for the best algorithm among ACS and EACS called ACS* with the latest best known solutions. Thus, column ACS* of Table 3 presents the average result, over five runs, of ACS* algorithm.
We can see from Tables 1 and 3 that on Solomon instances with 100 customers, HILS algorithm performs, on average, better than GVNS and DABC on Class 1 (C100, R100 and RC100) both with regard to solution quality and computational time. On C100 and R100 test instances, HILS algorithm performs, on average, better than FSA, GA and IterLS using approximately the same computational effort. On these instances of Class 1, ILS, SSA and GRASP-ELS and I3CH are, on average, better than HILS at the cost of extra computational time. On the test instances of Class 2 (C200, R200 and RC200), HILS algorithm performs, on average, better than GA using more computational time. On C200 test instances, HILS performs, on average, better than GVNS both with regard to solution quality and computational time. On these instances of Class 2, most of stateof-the-art algorithms outperform HILS algorithm using more computational effort. On these test instances of Solomon with 100 customers, HILS achieved the optimal solution 19 times.
On the test instances of Cordeau, HILS algorithm is not as competitive as state-ofthe-art algorithms on PR01-10 test instances; HILS algorithm achieved the worst average gap which is equal to 6.8 %. On PR11-20 test instances of Cordeau, HILS algorithm performs, on average, better than ACS* both with regard to solution quality and computational time and better than FSA using more computational effort. However, the rest of state-of-the-art algorithms perform, on average, better than HILS on these test instances of Cordeau. HILS algorithm is able to achieve 1 new best solution on PR17 test instance.
On test instances with 50 customers, HILS algorithm is as competitive as VAN algorithm both with regard to solution quality and computational time. VNS algorithm performs, on average, better than HILS and VAN algorithms at the expense of a reasonable amount of computational time. On these instances, HILS is able to achieve the optimal solution on 19 instances over 29.

Computational results on OPSTW instances
In this section we computationally test the increase of the profit due to soft time windows.   Table 4 presents the results obtained by HILS algorithm on OPSTW test instances. Column one (Instance) presents the instance over which the algorithm is tested. Column two corresponds to the maximum allowable time window violation and the maximum allowable waiting time at any customer. Columns three to six present the best, worst and average profit, over five runs, of HILS algorithm and the average computational time. Columns N R and %HTW present the number of routed customers on the average solution and the percentage of non-violated time windows on this solution respectively. The last column presents the sequence in which the customers are routed in the average solution.

Table 3 Overall comparison on OPTW test instances
The results show, as expected, that in all cases it is possible to increase the collected profit by allowing controlled violations of time windows. Allowing for example the maximum allowable violation of time window to 1 % of the maximum time duration and the maximum allowable waiting time at any customer to 7 % of the maximum time duration, results in solution with a profit of 330 and 95 % of non-violated time windows while the profit reported for hard time windows is 308. On the other hand, setting for example P max to 4.5 % and W max to 5 %, results in 23 routed customers while the number of routed customers reported for hard time windows is 21.