Genetic Algorithm Solving Orienteering Problem in Large Networks

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1 28 Advances in Knowledge-Based and Intelligent Information and Engineering Systems M. Graña et al. (Eds.) IOS Press, The authors and IOS Press. All rights reserved. doi: / Genetic Algorithm Solving Orienteering Problem in Large Networks Joanna Karbowska-Chilińska, Jolanta Koszelew, Krzysztof Ostrowski and Pawe l Zabielski Bialystok University of Technology, Faculty of Computer Science, Poland Abstract. This paper proposes an effective genetic algorithm (GA) with inserting as well as removing mutation so as to solve the Orienteering Problem (OP). In order to obtain improved results we have taken into consideration not only the total profit but also the travel length for the given path. The computer experiments that have been conducted on a large transport network (approximately 900 vertices) present better solutions in comparison to the well-known Guided Local Search (GLS) method. It should be stated that GA is significantly faster than GLS. 1 Introduction The Orienteering Problem (OP) can be modeled as a weighted and complete graph problem. Let G be a graph with n vertices, each vertex has a profit p i 0. Each edge between vertices i and j has cost t ij associated with it. The triangle inequality is satisfied for G. The starting point s and the ending point e are given. The objective is to find a path from s to e that maximizes the total profit. In addition, the total cost of the edges on this path should be limited by the given constrained t max and each point on the path is visited only once. The OP is applied to numerous solutions of real problems. One of the applications mentioned in the literature is a travelling salesman problem [Ts84]. In this case a salesman could visit a subset of customers, who urgently require a delivery of products without violating the travel-time distance. Another application of the OP is clearly seen in the products planning production [Ba89]. In this case the objective is to maximize the total profit of production without exceeding the production time constrains. The OP can also be used to model the tourist tripplanning problem [VSVO11], [SV10]. Usually tourists visiting any region are not able to visit all attractions because they are constrained not only by the time of the trip but also by money. Using the above stated solution, they would stand a chance of visiting the most valuable attractions in the presupposed time limit. The OP is NP-hard [GLV87], so exact algorithms are very time consuming. For practical applications the following meta-heuristic solutions are usually used instead: genetic algorithms[ts00], the local search methods [CGW96], [VSVO9], tabu search [TMO5] and the ant colony optimization approach [SS10]. Usually the meta-heuristic solutions presented in the literature, provide effective solutions in a reasonable time for only short routes [VSVO11] (e.g. considering

2 J. Karbowska-Chilińska et al. / Genetic Algorithm Solving Orienteering Problem 29 one city or one region of the given country). However, the real OP problems can also be modeled by using large networks. For this reason, the presented paper offers the effective genetic algorithm for the OP, which yields the high-quality solutions in large networks. Moreover, our method has been tested on the real transport network, which contains 908 cities all over the territory of Poland. During the tests the long routes were generated (even t max = 3000 km) inthe satisfactory time. The GA proposed in this article could be applied to the following exemplary situations. Given the fact that the score of each city is assigned basing on the interests of previous visitors, the proposed solution could significantly simplify the work of any travel agency in the case of planning the most attractive tour around the country (limited by the fixed number of kilometers). In another application, it can be observed that if the score of the city corresponds to the number of people living in that city, the most valuable concert tour for a group of artists in the country could be generated limited only by the fixed number of kilometers. The paper is organized as follows. Section 2 outlines an overview of the previous exact solutions and meta-heuristic approaches to the OP. Section 3 gives detailed specifications of the proposed GA with the special type inserting and removing mutation. Computational experiments and the comparison of the GA results with the GLS method [VSVO9] and the previous version of our genetic algorithm (pga) [OK11], [PK11] are discussed in Section 4. Conclusions and further work are drawn in Section 5. 2 Literature review It should be mentioned that two categories of research concerning the OP are widely considered: the exact solutions as well as the heuristic approaches. The exact algorithms solving the OP are based on the branch and bound and branch and cut methods. Fischetti et al. [FST98] has given the exact solution for the examples reaching up to 500 vertices and Gendreau et al.[gls98] has solved instances reaching up to 300 vertices through the usage of the branch and cut method. What is more, Ramesh et al. [RYK92] has developed the branch and bound framework within Lagrangean relaxation and has solved instances up to 150 control points. It was Tsiligrides [Ts84] who introduced the first heuristic approaches to solve the OP. His S algorithm is based on the Monte Carlo method where a lot of paths are generated and the route with the highest score is selected. During the construction of the route for points, which are not included on the route the probability of selection is assigned. The probability is based on the Euclidean distance (between the last point on the route and the analyzed vertex) and the score of the vertex under consideration. In the second method the search areas are divided into sectors that are determined by two concentric circles and the arc of a known length. In the next step, paths are built into separate sectors on the base of defined rules and the best one is selected.

3 30 J. Karbowska-Chilińska et al. / Genetic Algorithm Solving Orienteering Problem Golden et al. [GLV87] introduced the three-phased iterative heuristic, which consists of route construction using the greedy algorithm as well as the center of gravity. Route improvement using 2 -op procedure and center of gravity is computed again in the last step. Golden et al. [GWL88] worked out the improved iterative heuristic, which links Tsiligrides S-algorithm concept, the center of gravity and learning capabilities. Chao et al. [CGW96] introduced a two-step iterative heuristic, consisting of initialization and improvement steps. The method only considers vertices lying in the ellipse using t max as a length of the major axis. Several routes are generated with the help of the greedy algorithm and the one with the highest score is chosen. In the improvement phase a two-point exchange in the cheapest way is applied and next, improvement point is used in order to increase the total score. Finally, 2-op procedure is applied to decrease the total length of solution and when the t max is exceeded, the certain numbers of vertices with low score are removed and improvement phase is repeated. Tasgetiren et al. [TS00] worked out the first genetic algorithm for the OP. The tournament selection, the injection crossover as well as the mutation using elements of the local search method (add, omit, replace and swap operators) were created by them. They also proposed the penalty function, which helped to create the infeasible solution. Vansteenwegen et al. [VSVO11] developed the guided local search method (GLS) for the OP, which compares the mentioned method reducing the chances of trapping in the local optimum. For each local search iteration, they used special penalties e.g. increase of the score of the non-included locations and decrease of the score of the included location. First, some of the initial routes are generated on the basis of the Chao method [CGW96] and the best one is selected. Every local search iteration consists of three phases. In the first TSP phase the 2-opt procedure is used to reduce the length of the route. In the second phase nonincluded locations are inserted in the tour in place of the least consuming travel length. The order in which locations are considered in case of the insertion is dependent on the special ranking. In the third phase all non-included locations are checked one by one. If remaining budget is insufficient to include the considered location, all included locations with the lower scores are considered for exclusion and as a result they are replaced by the non-included node. If no replacements are possible, the heuristic reaches a local optimum and the penalty function is applied. Next, replacements are repeated a fixed number of times and the local search iteration of the diversification procedure is applied in order to explore the whole solution space. The GLS meta-heuristic method gives satisfactory results for mediocre networks and can be found in the Mobile Tourist Guide [VSVO11], [So10]. Solutions mentioned in the literature were usually tested in small networks (the number of vertices in Tsiligirides [Ts84], Chao et al. [CGW96] and Fischetti et al. [FST98] benchmark instances vary between 21 and 500). In practice networks modeling the OP problems could be significantly larger (more than 500 vertices). This article confirms that the solution described in the next section

4 J. Karbowska-Chilińska et al. / Genetic Algorithm Solving Orienteering Problem 31 may be recommended for large networks as it provides better results than the GLS. 3 Genetic Algorithm Route determination in GA is a multi-stage process involving the following steps: 1. Initialization: First, the initial population of P size solutions is generated. Tours are encoded into chromosomes as a sequence of vertices (cities). It is the most natural way of adopting GA for OP. At the beginning a random vertex v, which is adjacent to vertex s (the start point) is chosen. Next, the value t sv is added to the current travel length. If the current travel length does not exceed 0.5 t max, the tour generation is continued. Now we start at vertex v and choose a random vertex u adjacent to v. At every step we exclude the previously visited vertex from the set of possible to choose from vertices. This action prevents from visiting the given vertex repeatedly. If the current tour length is greater than 0.5 t max, the last vertex is rejected and we return to vertex s the same way but in the reversed order. This way of generating the individuals of the initial populations means that they are symmetrical in respect to the middle vertex in the tour. However, these symmetries are removed by the algorithm. In tab. 2 the example of the initial population for P size =5 and t max =30 is presented for the graph defined by the matrix of distances and profits introduced in tab. 1. Table 1. Graph G example vertex profit Table 2. Initial population example (P size=5, t max=30) no Individual travel length total profit fitness 1 (1, 2, 3, 4, 3, 2, 1) (1, 5, 4, 3, 4, 5, 1) (1, 4, 3, 4, 1) (1, 5, 3, 5, 1) (1, 4, 5, 4, 1)

5 32 J. Karbowska-Chilińska et al. / Genetic Algorithm Solving Orienteering Problem 2. Evaluation: We follow to calculate the value of the fitness function F to evaluate the optimum nature for each route (chromosome) included in P. The fitness function should estimate the quality of individuals, according to the sum of profits T otalp rofit and the length of the tour T ravellength. Each vertex can be visited more than once in the route but the T otalp rofit is increased when the vertex is visited for the first time. In the GA the fitness function is equal to T otalp rofit 3 /T ravellength. This variant of the fitness function gives the best results from among all the tested versions, so that, the final population contains a route with the highest total profit. 3. Improving population: After generating initial population, the GA starts to improve the current population through repetitive application of selection, crossover and mutation. The algorithm stops after n g generations and the result tour is the best individual (with the highest F ) from the final generation Tournament selection: In the first step, tournament selection is applied - we select t size random, numerous individuals from the current population, and the best one from the group is copied to the next population. It is important that we choose the best chromosome from the group that is the one with the highest value of T otalp rofit 3 /T ravellength. Next, the whole tournament group is returned to the old population, and finally after P size repetitions of this step a new population is created Crossover: The crossover is performed as follows: First two random individuals are selected - parents. Afterwards we randomly choose a common gene (crossing point) from both parents (the first and the last genes are not considered). If there are no common genes, crossover cannot be done and no changes to the chosen chromosomes are applied. Following this, two new individuals are created as a result of exchanging chromosome fragments (from the crossing point to the end of the chromosome) from both parents. If one of the children does not preserve t max constraint, the fitter parent from the new population replaces it. If both children do not preserve this constraint, the parents replace them in the new population (no changes applied). In the example presented in fig. 1 two children with low enough travel length are created. Parents can be symmetrical, however the crossover removes the symmetry from the offspring individuals Mutation: After the selection and crossover phases, the population undergoes the heuristic mutation. First, a random individual is selected to be mutated. There are two possible kinds of mutation: inserting a new gene or removing an existing gene. The standard version of the inserting mutation, in which the new gene is inserted in a random place of the individual, does not cause good enough improvement of the fitness function value. Therefore we apply the inserting mutation in which all possibilities of inserting a new available gene u that is not present in the chromosome are considered. The place with the highest value of p 2 u/t ravellengthincrease(u) (T ravellengthincrease(u) - increase of the travel length after adding the gene u to the chromosome) is cho-

6 J. Karbowska-Chilińska et al. / Genetic Algorithm Solving Orienteering Problem 33 sen. If T ravellengthincrease(u) is less than 1, we must consider the highest value of p 2 u. In the example presented in fig. 2 there are four possibilities of inserting a new gene into the chromosome (1, 4, 3, 5, 1) without exceeding t max but the insertion of gene 2 between genes 1 and 4 is chosen because this position of gene 2 gives the highest value of p 2 u/t ravellengthincrease(u) (202). The gene u is not available when the addition to the route exceeds t max or do not increase the value of currently best fitness function. In the removing mutation, only genes that appear in the chromosome more than once are considered (except for the first and last genes). If there are no such candidates the removing mutation is not performed (no fitness loss occurs). Otherwise, we choose the gene in order to shorten the travel length as much as possible. In the example presented in fig. 3 we have only one candidate suitable for removal, gene 5 (between genes 1 and 3 or between genes 3 and 1). If we remove this gene from the first position, the path is shorten by 6, but if we remove gene 5 from the second position (between genes 3 and 1) the path is shorten by 5. Thus, the best candidate to remove is gene 5 between genes 1 and Determining results: The route with the highest value of the fitness function F is the result of GA. Fig. 1. Crossover example (t max = 30), the crossing point is red The presented method is an improved version of the algorithm described in [OK11], [PK11]. The main difference between GA and its previous version, called

7 34 J. Karbowska-Chilińska et al. / Genetic Algorithm Solving Orienteering Problem Fig. 2. Inserting mutation (t max = 30 ), the inserted gene is red Fig. 3. Removing mutation (t max = 30), the removing gene is red farther pga, is clearly visible in the evaluation step. The fitness function F for GA takes into account not only the total profit but also travel length for the given path. Moreover, in GA the inserting mutation is far more effective and has significantly shorter execution time. In the previous version of this method, all possible (not only available as in GA) vertices were checked and the vertex u with the highest profit (not with the heuristics p 2 u/t ravellengthincrease(u) like in GA) was chosen. Moreover, the method pga presented in [OK11], [PK11] solves some modified version of the OP in which the vertices can be repeated in the final route and the transport network is not a complete graph. GA works on the standard version of the OP assuming that the given graph satisfies the triangle inequality. 4 Computational Experiment In this section results of GA and the GLS methods as well as pga are compared. The experiments have been carried out on the transport network, which contains 908 cities all over the territory of Poland. Each city is described by its longitude, latitude as well as its profit. The distances between any c 1 and c 2 cities are calculated (in meters) as follows: distance c1c 2 = arccos(sin(lat c1 ) sin(lat c2 )+cos(lat c1 ) cos(lat c2 ) cos(long c1 long c2 ))

8 J. Karbowska-Chilińska et al. / Genetic Algorithm Solving Orienteering Problem 35 where lat c1, lat c2, long c1, long c2 denote latitudes and longitudes of the cities (which are given in radians). Profit associated with a given city is determined according to the rule: the more inhabitants, the higher profit associated with the given city. Profit of each city is equal to the number of inhabitants divided by 1000 (profit is between 1 and 1720). The capital city of Poland, Warsaw, is established as the starting and the ending point. In GA, pga and GLS possible distance limits are 500, 1000, 1500, 2000, 2500, 3000 kilometers. The GA and pga parameters are determined experimentally and have the same values. Table 3 overviews the parameters and their values. To obtain statistics of GA and pga the route with the best fitness results from 30 algorithm runs was taken into account. In the experiments GLS parameters were the same as in [VSVO9] with the exception of the replace phase. In this step, the maximum number of iterations was determined by tests and it was equal to 700. Increase of this parameter (above 700) did not improve the quality of GLS results and moreover, it was time consuming. The experiments have been performed on implementations of GA, pga and GLS that run on the Intel Core 2 duo 2.8GHz CPU. The comparison of the name explanation Table 3. GA and pga parametres value P size initial population size 300 t size number of individuals chosen in a tournament selection 5 n g number of GA generations 1000 best solutions and the execution time according to the GA as well as the pga with the ones obtained from GLS is outlined in tab. 4. Results indicate that GA gives better profits. The percentage difference between GA and GLS profits varies from approximately 2% for t max = 2500 to 12% for t max = Only in the case t max = 1500, GA gives slightly worse profits (0.81%) than GLS, but GA is significantly faster. The difference between GA and pga profits varies from approximately 1.8% for t max = 3000 to 8% for t max = 2500 (only in case t max = 2000, pga gives better profit result (about 0.16%) than GA). The execution time of GA and pga is calculated as a sum from 30 runs. It is easy to see that the execution time of GLS is even more than three times higher than GA (e.g. for t max = 1500) but the execution time ofgaisevenmore six time lower than pga (for t max = 2000). It is interesting that in the algorithms, final paths have comparable lengths, which are very close to t max but the generated paths are usually different (passing through various cities). In fig. 4, examples of the best-generated routes are marked on the Polish map for GA and GLS. The last remark concerns the convergence of GA algorithm. The maximum number of iterations equals to 1000 but the best routes are generated earlier. As can be observed in fig. 5 the best route e.g. for t max = 3000 or t max = 1000 is generated

9 36 J. Karbowska-Chilińska et al. / Genetic Algorithm Solving Orienteering Problem after 500 and 250 iterations respectively. Every 100 generations GA checks the current population and stops if no improvements have been found. Table 4. GA, GLS and pga best results and execution time (time) comparison GA GLS pga t max profit length time profit length time profit length time GA GLS Fig. 4. Examples of routes generated for T max = Conclusion and Further Work Computer experiments have shown that GA performs much better than the comparable methods - GLS and pga. It is very important that both algorithms have been tested on a large network (about 900 vertices) with the high value of t max. GLS and other effective solutions for the OP were previously tested only on benchmark graphs with a much smaller number of vertices (between 21 and 500) [Ts84], [CGW96], [FST98]. On the other hand, it was found that large network applications require the usage of effective solutions of the OP. Therefore

10 J. Karbowska-Chilińska et al. / Genetic Algorithm Solving Orienteering Problem 37 Fig. 5. Profit of the best route from n-th generation GA research on effective method of solving the OP for the case of large networks is such a crucial issue. In addition, the GA is incomparably better than the GLS and pga methods in execution time. In the future work, it is intended to expand experimentation with other instances of large networks. If tests show poor performance of GA the new heuristics must be added to the algorithm. We also plan to improve the algorithm in the generation of the initial population step. The current version of GA contains only symmetric routes. We want to replace parts of symmetric routes by asymmetric ones. Dealing with this kind of routes also requires paying close attention to the topology of networks. Finally, we will also work on improving the crossover operator and reducing the execution time using our method. References [Ba89] Balas, E.: The prize collecting traveling salesman problem. Networks. 19 (1989) [CGW96] Chao, I. M., Golden, B. L., Wasil, E. A.: A Fast and effective heuristic for the orienteering. European Journal of Operational Research. 88 (1996) [FST98] Fischetti, M., Salazar, J. J., Toth, P.: Solving the Orienteering Problem through Branch-and-Cut. INFORMS Journal on Computing. 10 (1998) [GLS98] Gendreau, M., Laporte, G., Semet, F.: A branch-and-cut algorithm for the undirected selective traveling salesman problem. Networks. 32(4) (1998) [GLV87] Golden, B., Levy, L., Vohra, R.: The orienteering problem. Naval Research Logistics 34 (1987) [GWL88] Golden, B., Wang, Q., Liu, L.: A multifaceted heuristic for the orienteering problem. Naval Research Logistics 35 (1988) [OK11] Ostrowski, K., Koszelew, J.: The comparision of genetic algorithm which solve Orienteering Problem using complete an incomplete graph, Zeszyty Naukowe, Politechnika Bialostocka, Informatyka 8 (2011), [PK11] Piwonska, A., Koszelew, J.: A memetic algorithm for a tour planning in the selective travelling salesman, Foundations of Intelligent system, Springer-Verlag, Lecture Notes in Artificial Intelligence, 6804 (2011),

11 38 J. Karbowska-Chilińska et al. / Genetic Algorithm Solving Orienteering Problem [RYK92] Ramesh, R., Yoon, Y., Karwan, M.H.: An Optimal Algorithm for the Orienteering Tour Problem. INFORMS Journal on Computing. 4(2) (1992) [SS10] Sevkli, Z., Sevilgen, E.: Discrete particle swarm optimization for the orienteering Problem. IEEE Congress (2010) [So10] Souffriau,W.: Automated Tourist Decision Support, PhD Thesis, Katholieke Universiteit Leuven (2010) [SV10] Souffriau, W., Vansteenwegen, P.: Tourist trip planning functionalities: Stateof-the-Art and Future. ICWE Workshops (2010) [TMO5] Tang, H., Miller-Hooks, E.:A tabu search heuristic for the team orienteering [Ts84] problem. Comput. Oper. Res. 32(6) (2005), Tsiligirides, T.: Heuristic methods applied to orienteering. Journal of the Operational Research Society. 35(9) (1984) [TS00] Tasgetiren, M. F., Smith, A.E.: A genetic algorithm for the orienteering problem. Proceedings of the 2000 Congress on Evolutionary Computation San Diego. 2 (2000) [VSVO9] Vansteenwegen, P., Souffriau, W., Vanden Berghe, G., Van Oudheusden, D.:A guided local search metaheuristic for the team orienteering problem. European Journal of Operational Research.196 (2009) [VSVO9a] Vansteenwegen, P., Souffriau, W., Vanden Berghe, G., Van Oudheusden, D.: Iterated local search for the team orienteering problem with time windows. Computers O.R. 36 (2009) [VSVO11] Vansteenwegen, P., Souffriau, W., Vanden Berghe, G., Van Oudheusden, D.: The City Trip Planner: An expert system for tourists. Expert Systems with Applications 38(6) (2011) This work was supported by Bialystok University of Technology under grants S/WI/1/2011 and S/WI/2/2008.