A Randomized Population Constructive Heuristic for the Team Orienteering Problem

: The NP-hard (complete) team orienteering problem is a particular vehicle routing problem with the aim of maximizing the profits gained from visiting control points without exceeding a travel cost limit. The team orienteering problem has a number of applications in several fields such as athlete recruiting, technician routing and tourist trip. Therefore, solving optimally the team orienteering problem would play a major role in logistic management. In this study, a novel randomized population constructive heuristic is introduced. This heuristic constructs a diversified initial population for population-based metaheuristics. The heuristics proved its efficiency. Indeed, experiments conducted on the well-known benchmarks of the team orienteering problem show that the initial population constructed by the presented heuristic wraps the best-known solution for 131 benchmarks and good solutions for a great number of benchmarks.


Introduction
The Orienteering Problem (OP) was first introduced by (Golden et al., 1987). It originates from orienteering sport. According to (Zettam and Elbenani, 2016) the Orienteering sport is defined as an outdoor sport, played in heavily forested and mountainous areas. A set of control points are located in the forest. Each control point is associated with a score. Assuming that the start control point and the end control point are fixed. Competitors equipped with compass and map are required to visit a subset of control points starting from the start control point and ending at the end control point with the aim of maximizing their total score within a predefined amount of time. Several variants of OP has been described in literature such as The Team Orienteering Problem, the Orienteering Problem with Time Windows, the team orienteering problem with time windows and the time-dependent orienteering Problem. OP and its variant were given a great interest as results of their applications (Vansteenwegen et al., 2011). Selective travelling salesperson problem (Tsiligirides, 1984), home fuel delivery problem (Golden et al., 1987), single-ring design problem (Thomadsen and Stidsen, 2003) and mobile tourist guide problem (Souffriau et al., 2008) are some of well known application of OP and its variants. For more details readers are referred to (Vansteenwegen et al., 2011).
The team orienteering problem TOP extends OP. It was first introduced by (Chao et al., 1996). Competitors are subdivided into teams. The competitors of a given team collaborate to maximize the score within a predefined amount of time or distance limit. Each control point is visited once by a member of a given team. Even though two exact methods have been proposed to solve the team orienteering problem, it still be considered as an NP-hard (complete) problem according to (Chao et al., 1996). Therefore, a number of heuristics and metaheuristics have been developed to solve TOP. Lin (2013) proposed a multi-start simulated annealing algorithm that combines the simulated annealing algorithm and the multi-start hill climbing algorithm. Butt and Cavalier (1994) presented a mixed integer programming solved by a greedy method that adds the best pair of vertices to the solution tours at each iteration. They claims that their approach do well with relatively small number of control points within a tour. In the approach proposed by (Chao et al., 1996), the initial solution is constructed by inserting Control points into paths using the cheapest insertion rule. If unsigned control points remain new paths are constructed. The constructed initial solution is improved by using 1-point movement, 2-point exchange and 2-opt operator with record-to-record framework. Tang and Miller-Hook (2005) proposed a Tabu search embedded with adoptive memory. This method generates better solution with more computational time compared to (Chao et al., 1996) for a number of instances. (Archetti et al., 2007) addressed a variable neighborhood approach which outperforms (Tang and Miller-Hooks, 2005;Chao et al., 1996) approaches. Ke et al. (2008) proposed an Ant Colony Approach that embedded one of the three following methods: Sequential deterministic-concurrent method, randomconcurrent method and simultaneous method. The performance of the Ant Colony Approach with the sequential method is comparable to (Archetti et al., 2007). Vansteenwegen et al. (2009) introduced the disturb method which combines a guided local search method and a diversifying mechanism. The disturb method find the best-known solutions in shorter computational time compared to the existing methods. (Bouly et al., 2009) proposed a PSO-inspired algorithm which updated the best-known solution for one instance use this style when you need to begin a new paragraph.
In this study, we propose a randomized population constructive heuristic for the Team orienteering problem. This heuristic constructs a diversified initial population for population-based metaheuristics. It constructs solutions based on randomly generated permutations of control points. The solutions within the initial population are diversified with different costs and tour lengths. A number of permutations are randomly generated. For each permutation of control points a given number of path beginning from the starting control point and ending at the arrival control point are constructed. The best solution of the population is then enhanced by a local search.
The rest of this paper is organized as follows. The first section addresses the team orienteering problem. The first section also introduces the mathematical model adopted in this study. The second section details the proposed heuristic. The fourth section involves the computational results followed by concluding remarks and proposal for future works in the fifth section.

Mathematical Formulation
Given a set of n control points and m competitors. The control points are usually named locations and competitors are usually named tours in the literature of TOP. The main goal of TOP is to construct m tours starting from the departure (location 1) and ending at the arrival (location n) that maximize the total score denoted s. A travel time or length of a tour cannot exceed T max . In the present paper, we employ the Euclidean distance to calculate the length tours. Table   1 shows an example of TOP instances where T max = 20, m = 2. Table 1 contains locations, XY coordinates and scores. The solution of the instance shown in Table 1 is represented in Fig. 1.
In this study, a mathematical model similar to the one used by Lin (2013) is described as follows: , {0,1}, i, j 1,..., n; 1,..., ijk ik x y k m ∈ ∀ = ∀ = Where: S i = Is the score associated to the ith location t ij = The length of the path starting at the ith location and ending at the jth location. The travel length cannot exceed T max V = Is the set of locations U = Is a subset of V x ijm = 1 = if, in tour m, a visit to location is followed by a visit to location, 0 otherwise y ik = 1 = if location is i visited on tour k, 0 otherwise s ik = The start time of the service at location i on tour k The objective function to maximize is represented by Equation 1. The constraint that all tours starts from location 1 and ends at location n is ensured by Equation 2. The connectivity of tours is maintained by Equation 3. Constraint (4) guarantees that every location is visited at most once. Constraint (5) guarantees that length limitation constraint is not violated for each tour. Constraint (6) excludes sub-tours. Constraint (7) states that the variables x and y are binary. The Novel Randomized Heuristic The heuristic randomly generates a predefined number of permutations. Solutions are then constructed on the basis of the generated permutations. Each solution is represented by m connected tours. Given a permutation of n elements. Each element represents a location. For each tour k, the heuristic seeks a permutation. If the ith location is not contained in other tours and if adding the ith location does not violate fifth constraint described in the mathematical model, the ith location is added to the tour kth. Otherwise, the heuristic goes to the next iteration and so forth. Finally the best solution of the population is enhanced via a local search. The local search used in this study employs the swap and the add/drop operators. The solution is enhanced 10 times with a neighbourhood size equal to the half of the number of customer in the first tour. Our heuristic is describes in details in the algorithm (1) below. dist(i,j) denotes the Euclidean distance between two location i and j.
Algorithm (1) Generate randomly l permutations for each permutation do Apply the local search to enhance the best solution. Output: population of feasible solutions The randomized heuristic reaches best-known solutions for 131 of benchmarks in a minor computational time. The randomized heuristic also generates average solutions for the majority of the reminder benchmarks. A good initial population would allow population based metaheuristics to find good solutions in an optimistic computational time with less computational efforts.

Results and Discussion
The proposed heuristic was implemented using JAVA language and was run on a PC with an Intel Core i5-2540 M 2.60 GHz processor. To evaluate the performance of the proposed heuristic, each benchmark of TOP was tested over ten trial and compared to the best-known solution. The benchmarks we used in this study are available at (https://www.hds.utc.fr/~moukrim). Table 2 summarizes the characteristics of problem benchmarks sets. The first number of a problem indicates the set number, e.g., p2.2, p2.3 and p2.4 belong to set 2. The coordinates and score of each location are identical for the instances belonging to a given problem set. The second number in the problem set indicates the number of tours, e.g., p2.3 means that there are 3 tours.
We ran our heuristic 20 times on each instance in order to select the best obtained solution. Table 3 to 23 contain the problem nomination, the best-known solution in the literature and also the best solution in the population we obtained by applying the proposed randomized heuristic. The third and fifth column of each table contain the results obtained by generating a population of 10× T max and 10 elements. The best-known solutions in bold are obtained by exacts methods. In this section, we use the best-known solutions provided by (Kim et al., 2013).    Table 3 contains the results obtained for data set 1 with 2 tours. Six best-known solutions are obtained for a population size equal to 10× T max , while only four bestknown solutions are obtained for a population size equal to 10. The results obtained for data set 1 with 3 tours are contained in Table 4. Eight best-known solutions are obtained for population size equal to 10× T max , while seven best-known solutions are obtained for a population size equal to 10. The reminder tables contain the results of the rest sets. The obtained results by population size equal to 10× T max are better than those obtained by a population size equal to 10.
The obtained results prove the efficiency of our heuristic. Indeed, a great number of solutions is near to the best-known solutions. Other obtained solutions reach the best-known solution in minor computational time. Those results represent a promising start for populationbased metaheuristics.

Conclusion
This paper, introduces a randomized population constructive that generates the initial population for the team orienteering problem. The obtained results promote the integration of the heuristic to built initial population for population-based metaheuristics. The use of permutations for the proposed heuristic avoids infeasible solutions. Moreover, dealing with permutations during the search process would facilitate the application of operators and avoid infeasible solutions. Indeed, for each permutation a unique solution is associated by applying only the extern loop of the proposed heuristic. In this study, the best solution of the population is enhanced by a local search using the swap and the add/drop operators. This component allows finding better solutions.

Ethics
The author confirms that this manuscript has not been published elsewhere and that no ethical issues are involved.