AN EFFICIENT COMPOSITE HEURISTIC FOR THE SYMMETRIC GENERALIZED TRAVELING SALESMAN PROBLEM

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1 AN EFFICIENT COMPOSITE HEURISTIC FOR THE SYMMETRIC GENERALIZED TRAVELING SALESMAN PROBLEM Jacques Renaud and Fayez F. Boctor Télé-Université, Université du Québec, Canada, and Université Laval, Canada. ABSTRACT The main purpose of this paper is to introduce a new composite heuristic for solving the generalized traveling salesman problem. The proposed heuristic is composed of three phases: the construction of an initial partial solution, the insertion of a node from each non-visited node-subset, and a solution improvement phase. We show that the heuristic performs very well on thirty six TSPLIB problems which have been solved to optimality by other researchers. We also propose some simple heuristics that can be used as basic blocks to construct more efficient composite heuristics. Keywords: Generalized traveling salesman problem, heuristics. 1- INTRODUCTION This paper deals with the problem known as the generalized traveling salesman problem (GTSP); a special case of the well known traveling salesman problem (TSP). In this case, the salesman must pass through a number of predefined subsets of customers, visiting at least one customer in each subset while minimizing the sum of traveling costs. Thus, in addition to determining the order in which the subsets should be visited, the salesman should choose the customer or the customers to be visited in each subset. The GTSP may be defined on either a directed graph or an undirected graph and may be either symmetric or asymmetric. The most studied version of this problem is the one where the salesman should visit exactly one customer in each subset. Obviously, even if it is not required explicitly, the optimal tour will visit only one customer in each subset when travel costs are symmetrical and satisfy the triangle inequality (see Laporte and Nobert 1983).

2 2 As pointed out by Fischetti et al (1994), the generalized traveling salesman problem is NP-hard since it reduces to the TSP when each customer subset contains exactly one customer (the proof for the NP-hardness of the TSP can be found in Garey and Johnson 1979). Also, the GTSP has many potential applications. Laporte et al (1995) showed that a variety of combinatorial optimization problems can be modeled as GTSPs. Among these problems are the location routing problems, some material-flow system design problems, the post-box collection problem and the arc routing problem (for more details about these problems see Laporte et al 1995). The GTSP was simultaneously introduced by Srivastava et al (1969), who addressed the symmetrical version of the problem and proposed a dynamic programming approach for its solution, and by Henry-Labordere (1969) who addressed the asymmetrical case. Saksena (1970) addressed both the symmetric and asymmetric cases and presented an application in the field of scheduling clients through welfare agencies. Several years later Laporte and Nobert (1983) and Laporte et al (1987) addressed respectively the symmetrical and the asymmetrical GTSP. In both cases the problem was formulated as an integer program and a branch and bound algorithm for its solution was developed. More recently the GTSP has been the subject of two Ph.D. theses by Noon (1988) and Sepehri (1991). Noon and Bean (1991) presented an optimal approach to solve the asymmetrical GTSP where the subsets of customers are mutually exclusive, it is impossible to travel from one customer to another one in the same subset (no intraset arcs) and the sub-tour must pass through each subset exactly once. They noted that forbidding intraset traveling and requiring subsets to be mutually exclusive is not very restrictive since, as was shown by Noon (1988), a problem with intraset arcs and overlapping sets can be transformed to a one with mutually exclusive sets and no intraset arcs. Noon and Bean (1993) showed how to transform the asymmetric GTSP into an equivalent clustered TSP. This latter problem is similar to the standard TSP except that the set of customers is partitioned into a number of clusters (subsets) and each cluster must be visited contiguously (i.e. the salesman cannot pass to a new cluster before visiting all the customers of the current cluster). Consequently we can solve the GTSP by using one of the clustered TSP solution techniques (see for example, Chisman 1975, Jongens and Volgenant 1985 or Gendreau et al 1994). Noon and Bean (1993) also showed how to construct, for any GTSP instance, an associated asymmetrical TSP instance such that the optimal solution of the GTSP can be drived straightforwardly from the optimal solution of the TSP. However, the associated

3 3 TSP is not equivalent to the original GTSP because a feasible (but not optimal) solution to the TSP may correspond to an infeasible solution of the GTSP. This implies that, for large problems which cannot be solved to optimality within a reasonable amount of computation time, we may not be able solve the GTSP by applying a heuristic to the associated TSP. Finally, Fischetti et al (1994) proposed a branch and cut algorithm to solve the symmetric GTSP to optimality. In this paper we propose a fast composite heuristic, called the GI 3 (Generalized Initialization, Insertion and Improvement) heuristic, to solve the GTSP. We also propose and study the performance of three other composite heuristics and provide some numerical results to prove the relative efficiency of the GI 3 heuristic. The remainder of the paper is organized as follows. In section 2 we point out that the GTSP is a variant of the traveling salesman subtour problem (TSSP) and review the literature on some other variants of the TSSP emphasizing the relationship between these variants. In section 3 we propose some simple heuristics to solve the GTSP. These simple heuristics can be used as basic blocks to construct more efficient composite ones such as the GI 3 and the three other composite heuristics proposed in this paper. In the fourth section we introduce the GI 3 heuristic and in the fifth section we provide the results of the evaluation study conducted to assess its performance. The conclusions are given in section SOME RELATED PROBLEMS The generalized traveling salesman problem is a special variant of the traveling salesman sub-tour problem. The TSSP is a traveling salesman problem where the salesman need not to visit every customer and may visit some of his customers more than once. Many other well known problems, such as the prize collecting traveling salesman problem, the multiobjective vending problem, the traveling salesman problem with non-visiting penalties, the time-constrained traveling salesman problem, the orienteering problem, the traveling purchaser problem, the covering salesman problem, the median tour problem and the maximal covering tour problem, can be viewed as particular cases of the traveling salesman sub-tour problem. The prize collecting traveling salesman problem (PCTSP) is the problem where a salesman travels from one customer to another at a given cost, pays a penalty for every customer he fails to visit and collects a prize for each customer he visits. The salesman's objective is to collect an amount of prize money greater than or equal to a given amount

4 4 while minimizing the sum of his travel costs and penalties to be paid. This problem was first introduced by Balas and Martin (1985). Balas (1989) discussed the structural properties of its polytope and Fischetti and Toth (1988) developed a branch and bound algorithm for its optimal solution. The multiobjective vending problem (MVP), introduced by Keller (1985), is a prize collecting traveling salesman problem that contains two objectives: maximizing the amount of prize money by visiting as many customers as possible and minimizing the sum of travel cost and non-visiting penalties. Assuming that these two objectives can not be aggregated in one, Keller discussed a number of possible approaches that could be used to derive the non-inferior solutions set. The traveling salesman problem with non-visiting penalties (NVP-TSP) can be viewed as a special case of the PCTSP where there is a penalty if the salesman fails to visit a customer but there are no prizes to collect. This problem was addressed by Volgenant and Jonker (1987) and they showed how to transform this problem into a regular TSP. Also, Laporte et al (1984) presented a branch and bound procedure to solve the traveling salesman sub-tour problem where some specified customers must be visited and the other customers may or may not be visited. This special version is called the traveling salesman sub-tour problem with specified nodes (SN-TSSP). It can be viewed as a special case of the NVP-TSP where the penalty of not visiting the specified customers is extremely high and cannot be afforded. The time-constrained traveling salesman problem (TCTSP) and the orienteering problem (OP) are two other prize collecting problems where traveling from one point to another requires some amount of travel time and the objective is to maximize the collected amount of prize money within a limited amount of total travel time. The difference between these two problems is that in the OP the starting and destination points are specified while in the TCTSP the traveling person may start from any point and has to come back to this same point. The TCTSP was first introduced and discussed by Cloonan (1966). Gensch (1978) presented one of its industrial applications and Golden et al (1981) developed a simple iterative procedure for its solution. On the other hand, the orienteering problem has received more attention. Several heuristics were developed (see for example Tsiligirides (1984), Golden et al (1987), Golden and Lui (1989), Keller (1989) and Ramesh and Brown (1991)) and Ramesh et al (1992) proposed an optimal solution approach. The traveling purchaser problem (TPP) is the problem where a purchaser should buy n different items while visiting some of the markets where these items are offered but at

5 5 different prices. The purchaser's goal is to buy all the required items while minimizing the sum of purchase and travel costs. Golden et al (1981) mentioned that this problem was first outlined in 1978 by Ramesh (in an unpublished manuscript) and they suggested a savings based heuristic for its solution. Another traveling salesman sub-tour problem is the covering salesman problem (CSP). Current and Schilling (1989) defined this problem as the one of identifying the minimum cost sub-tour such that every non-visited customer is within a predetermined covering distance of at least one visited customer. They suggested a mathematical formulation of the problem and presented a heuristic for its solution. Gendreau et al (1995) proposed a heuristic and a branch and cut algorithm to solve the CSP. Finally, Current and Schilling (1994) introduced two other bicriterion subtour problems: the median tour problem (MTP) and the maximal covering tour problem (MCTP). In both problems we have to identify p of n locations and to construct the shortest tour through the p selected locations. In addition, in the MTP, the second objective is to minimize the total demand weighted distance that the non visited locations should traverse to reach their nearest visited location. The second objective for the MCTP is to maximize the total demand that is within some prespecified maximal distance from a tour stop (visited location). 3- SOME SIMPLE HEURISTICS The symmetric generalized traveling salesman problem addressed in this paper can be defined as follows. Given an undirected graph G=(N,E) where the set of nodes N={1,..., n} is partitioned into m mutually exclusive and exhaustive subsets S k ; k=1,..., m, and to each edge (i,j) E, E={(i,j): i,j N, i j} is associated a distance (or cost) d ij such that d ij =d ji, determine the minimum-cost m-edge cycle on G which includes exactly one node from each node-subset. As mentioned above, Fischetti et al (1994) proposed a branch-and-cut algorithm to solve this problem to optimality. However, for very large problems we may not be able to obtain the optimal solution in a reasonable amount of computational time and consequently we may need a good heuristic method to solve such problems. Also, heuristics are needed if we have to solve the GTSP a large number of times. Heuristic methods can be divided into eight different categories (see Ball and Magazine 1981) named: construction, improvement, mathematical programming, decomposition, partitioning, restriction of feasible space, relaxation and composite heuristics. In this

6 section we present some simple decomposition, construction and improvement heuristics for the GTSP Decomposition heuristics We may decompose the GTSP into two sub-problems: the selection of nodes to be visited and the construction of a tour that visits the selected nodes. We call this kind of decomposition heuristics "node selection and tour construction heuristics". They consist of choosing a set of nodes C which contains exactly one node of each node-subset and then use a TSP method to determine the minimum or a near minimum Hamilton cycle on C. Obviously, the quality of the solutions we may obtain by this kind of approach depends heavily on the way we select the nodes of C and we do not expect that this approach could lead to the best results. A second decomposition approach starts by determining the order in which the m nodesubsets should be visited and then constructs the required m-edge cycle by determining the shortest tour visiting the node subsets in the chosen order. We call this kind of decomposition heuristics "subsets ordering and tour construction heuristics". The shortest tour problem can be solved in polynomial time by applying the following algorithm, called hereafter the ST algorithm. Assume that the node sets are renumbered by giving the number 1 to any of them and numbering the other sets according to the chosen visiting order. Let L ij denotes the of the shortest path linking a node i belonging to S 1 to a node j belonging to S k while passing by only one node in each of the sets S 2,..., S k-1. Also let L i be the of the shortest m-edge cycle starting from a node i belonging to S 1 and coming back to this same node. Then L, the of the shortest tour visiting each of the node sets exactly once in the chosen order, can be calculated as follows: L = min S L i, i 1 where: Li = min ij ji j S L + d ; i S1, m Lij = min Lil + dlj ; i S1, j S k, k > 2, l S k 1 and Lij = dij ; i S1, j S 2. The time complexity of this shortest tour dynamic programming algorithm is O(n 3 /m 3 ). However we can significantly reduce its computation time by choosing as S 1 the nodesubset containing the smallest number of nodes.

7 7 Several methods can be used to choose the order (sequence) for visiting the m nodesubsets. For example, we may argue that in some cases a good tour which visits all the n nodes indicates a good order for visiting the node-subsets. This is the case if the intrasubset distances are relatively small with respect to the inter-subset distances. Consequently, we may suggest to start by using a good TSP solution method to construct a tour that visits all the n nodes and then try to deduce from this solution the required visiting order. In this paper we propose and study the performance of the following order determination method, called the MO method (Multiple Order method). First we use the I 3 heuristic (see Renaud et al 1996) to construct a tour that visits all the n nodes. A subset visiting order can then be deduced as follows. Starting with the first node in the n-node tour, record its subset number as the first subset in the visiting order. Consider the next node in the tour and add its subset number to the visiting order if and only if it is not yet added. Repeat until all the n-nodes are considered. The MO method repeats this procedure n/10 times starting each time from a different node, uses the ST algorithm to determine the shortest tour for each visiting order, and retains the best solution. The starting nodes are nodes 10, 20, 30,..., 10 n/10. Obviously this method may generate the same order more than once. Thus the MO method compares each order to the previously generated orders and drops it if it is identical to a previously generated one. The complexity of the MO method is a function of the complexity of both the I 3 heuristic (Renaud et al 1996) and the ST algorithm. But as we can not determine the complexity of the I 3 heuristics (because we can not predict the number of times its improvement procedure will be executed) we can not give the exact complexity of the MO method Construction heuristics Construction heuristics consist of simultaneously choosing the nodes to be visited and constructing the required m-edge cycle. Among the methods belonging to this approach are some adaptations of some known TSP tour construction heuristics. For example Noon (1988) suggested adapting the nearest neighbor heuristic as follows. Starting with a given node, the final m-edge tour is constructed by adding one node at a time and the node to be added is the nearest node, among those belonging to non-visited node-subsets, to the last added node. This procedure is repeated n times starting each time with a different node and the best of the resulting n solutions is retained. Consequently the complexity of this adaptation of the nearest neighbor heuristic is O(mn 2 ). Noon suggested also five other heuristics and, based on a number of randomly generated test-problems, concluded that this adaptation of the nearest neighbor heuristic, called hereafter the NN heuristic,

8 8 outperforms all the other five. Fischetti et al (1994) proposed an adaptation of the farthest insertion heuristic and suggested that both the nearest insertion and the cheapest insertion heuristics can be adapted in a similar way. In this paper we propose and study the performance of an another construction heuristic. As this heuristic uses the cheapest insertion criterion it will be called the CI heuristic. It starts by identifying, in each node-subset S k, the node having the minimum sum of outgoing distances to nodes not belonging to S k. Then it applies the CLOCK heuristic proposed in Renaud et al (1996) on the set of identified nodes in order to construct an enveloping sub-tour (we give more details about CLOCK in the next section). Often, the enveloping sub-tour contains some but not all the identified nodes; however, it does not contain more than one node from any node-subset. To construct a complete GTSP solution we should insert into this sub-tour exactly one node from each non-visited nodesubset. This can be achieved by repeating the following steps until there are no more nonvisited subsets. Calculate the additional cost of inserting each node belonging to the nonvisited subsets and insert the node having the cheapest insertion cost. The complexity of the first step of the CI heuristic (the identification of the nodes having the minimum sum of outgoing distances) is O(n 2 ), the complexity of the second step (the application of the CLOCK heuristic on the set of m nodes having the minimum sum of outgoing distances) is O(m log m) and the complexity of the third step (the insertion step) is O(mn log n). Consequently the overall complexity of the CI heuristic is either O(n 2 ) or O(mn log n) depending on the values of m and n Solution improvement heuristics Any of the above methods can be followed by a solution improvement procedure. The simplest of these procedures is to use a standard TSP tour improvement procedure, like the 2-opt and 3-opt heuristics (Lin 1965) or the 4-opt* heuristic (Boctor and Renaud 1993 and Renaud et al 1996), to improve the GTSP sub-tour. Such a procedure may improve the sub-tour by changing the order in which the sub-tour nodes are visited but will never change the set of visited nodes. A better approach is to use an improvement procedure which is also able to modify the set of visited nodes if this can lead to an additional improvement. Hereafter we suggest three adaptations of Lin's heuristics called G2-opt, G3-opt and G- opt heuristics. Fischetti et al (1994) provide another improvement heuristic, called RP1, based on Lin's 2-opt and 3-opt heuristics. The G2-opt (generalized 2-opt) heuristic which

9 9 can be viewed as an adaptation of the standard 2-opt heuristic is as follows. Let S a,..., S e, S f, S g,..., S p, S q, S r,..., S x indicate the subset visiting sequence (order) according to the current GTSP solution (see Figure 1). The G2-opt heuristic chooses any part of this sequence, say S f, S g,..., S p, S q, and reverses it. This produces the sequence S a,..., S e, S q, S p,..., S g, S f, S r,..., S x. Then we use the ST procedure to determine the shortest m-edge cycle which visits the node-subsets in the new sequence and if this new cycle is better than the previous one, we choose it to be the current solution. A complete iteration of this G2-opt heuristic consists of trying all the possible sequence parts of the current sequence and the heuristic stops if no improvement can be achieved during a complete iteration. If we consider the ST procedure as an elementary operation, the time complexity of a complete iteration of the G2-opt heuristic is O(m 2 ). But, as the complexity of the ST heuristic is O(n 3 /m 3 ), the overall complexity of one G2-opt iteration is O(n 3 /m). Unfortunately, we can not determine the complexity of the whole procedure as we can not predict the number of iterations needed to reach the stopping condition. The standard 3-opt heuristic can be adapted in a similar way to produce the G3-opt heuristic and we can see that the complexity of one complete iteration of the resulting heuristic is O(n 3 ). Finally, the G-opt improvement heuristic is a restricted version of the G3-opt heuristic in which the sequence improvement move is to change the position of only one node subset. The ST heuristic is then used to determine the shortest m-edge cycle corresponding to the new sequence. A complete iteration of the G-opt heuristic consists of considering all the node sets and, for each node set, considering all the possible positions for its insertion. Thus the time complexity of a complete iteration is O(n 3 /m).

10 10 S a S x S e S r S f S q S g S p Current Solution Modified Solution Figure 1: Sequence modification according to the G2-opt heuristic 4- GI3: A NEW COMPOSITE HEURISTIC Hereafter we propose a new composite heuristic, called GI 3 (Generalized Initialization, Insertion and Improvement), to solve the generalized traveling salesman problem. This heuristic, which can be viewed as a generalization of the heuristic I 3 proposed in Renaud et al (1996), is composed of three phases: Initial sub-tour construction phase, Insertion phase, and Improvement phase. In the next section we analyze its performance and provide some computational results to assess its efficiency. It will be shown that based on the thirty six TSPLIB problems solved by Fischetti et al (1994), the GI 3 solutions are in average within 0.98 % of the Phase 1: Construction of an initial sub-tour The first step of this phase is to calculate for each node set S k and for each node i S k, the sum of outgoing distances linking i to nodes in the other subsets, denoted d i,where : di = d ij ; i S k, k = 1,..., m, j S k and to find for each set S k the node i k S k which has the smallest sum of outgoing distances; thus : d i k = min. i i S d k

11 11 The remainder of this phase consists of applying the sub-tour construction procedure called CLOCK (see Renaud et al 1996) on the set of nodes Cmin={ik ; k=1,..., m}. This procedure produces an enveloping sub-tour which contains some but not necessarily all of the nodes of Cmin. This envelope is defined by an ordered set of nodes, denoted H. The first node to be inserted in H is the northernmost node among the nodes of Cmin. In case of a tie, CLOCK chooses the easternmost node among those northernmost nodes. The first step of CLOCK can be stated as follows. Repeat until there are no nodes to the east of the last inserted node: Insert the northernmost node (among the nodes not included in H) to the east of the last inserted node. In case of a tie, choose, among the northernmost nodes, the easternmost one. The second step is identical except that we look for the easternmost node to the south of the last inserted node. In the third and forth steps we look respectively for the southernmost node to the east and the westernmost node to the north of the last inserted node. The envelope is closed when we return to the first inserted node. The complexity of the first step of this phase (identification of the nodes having the minimum sum of outgoing distances) is O(n2) and the complexity of the second step (application of the CLOCK heuristic on the set Cmin) is O(m log m). Thus the overall complexity of this phase is O(n2). Obviously, CLOCK applies to symmetric GTSPs defined on a plane in which each node is represented by its coordinates (xi,yi). In cases where we do not dispose of such coordinates we suggest using as initial sub-tour the one composed of the 3 nodes belonging to Cmin that are farthest from each other Phase 2: insertion of the remaining node sets The objective of this phase is to construct a sub-tour which visits exactly one node in each node set. Thus this phase is not executed if the sub-tour H visits all the nodes of the set Cmin. Let H visit the node sets Sa,..., Sd, Se, Sf, Sg,..., Sz in the indicated order and let the corresponding visited nodes be ia,..., id, ie, if, ig,..., iz. For each non-visited node set Sk and for each node i Sk, we calculate the additional cost resulting from inserting i between any two consecutive sets in H, say Se and Sf (notice that to keep the tour cost as low as possible we may need to change the nodes originally visited in Se and/or Sf). This additional cost, denoted deif is given by: d eif = Se S f min { di j d ji dil d li} { dii dii d d g d e e f ii} f g j, l.

12 12 Then the node to be inserted and the position for its insertion are those corresponding to the smallest additional cost. To improve this insertion procedure, after each insertion, the G3-opt procedure is applied to the chain of node sets containing, in addition to the inserted node set, its r predecessors and its r successors. The value of r is a parameter to be chosen by the user. Our experiment, reported in section 5, suggests to use r=5. However, the user may try to solve his problem several times with different values of r and retain the best solution. We can not determine the complexity of this phase as we can not predict the number of iterations of the G3-opt needed after each insertion Phase 3: Solution improvement The solution procedure we suggest is composed of two steps. In the first we apply the G- opt improvement procedure and in the second we apply the G2-opt procedure. These two improvement procedures were described in the previous section. Again we can not determine the complexity of this phase as we can not determine the exact complexity of the G-opt and G2-opt improvement procedures. Thus the exact complexity of GI3 can not be determined. 5- COMPUTATIONAL RESULTS To evaluate its performance, the GI3 heuristic was compared to three other composite heuristics all composed of an initial solution construction phase followed by the application of the G-opt and the G2-opt improvement heuristics. The heuristics used to construct an initial solution are the NN, the CI and the MO heuristics presented in section 3. However, as the MO heuristic generates several solutions, the improvement heuristics are applied to each of these solutions and we retain the best improved one. Notice that we can not determine the exact complexity these three composite heuristics as they all use the G-opt and G2-opt improvement heuristics. All the four composite heuristics were tested on 36 problems drawn from Reinelt's (1991) TSPLIB library of test problems and solved to optimality by Fischetti et al (1994) who developed a branch and cut solution procedure. For these problems, they reported an average solution time (on a HP 9000 series 700 Apollo) of seconds, a minimum time of 1.9 seconds and a maximum time of seconds. Originally the set of nodes in these problems are not divided into node-subsets. To divide them into subsets we used the procedure called CLUSTERING proposed by Fischetti et

13 13 al (1994). This procedure sets the number of subsets m= n/5, identifies the m farthest nodes from each other, called centers, and assigns each remaining node to its nearest center. Obviously, some real-world problems may have different cluster structures. However the solution procedures presented in this paper are able to handle any cluster structure. Prior to this comparison we carried out preliminary tests to determine a good value for the parameter r, the local optimization parameter (see phase 2 of GI3). Table 1 gives the average percentage increase over the optimal solution and computing times (in seconds on a Sun Sparc Station LX) for the 36 test-problems. As expected computing times increases with r, however, when a post optimization procedure was used, no improvement was achieved for r>5. Table 1: Average percentage increase over the for different values of r Value of No post optimization After the G-opt heuristic After G-opt and G2-opt r % increase Time % increase Time % increase Time To asses the performance of the first two phases of the proposed heuristic alone, called hereafter GI2 (Generalized Initialization and Insertion), we compared their results to those obtained by the NN, the CI, and the MO heuristics without any post-optimization. Table 2 gives the results of this comparison. From this table we see that GI2 largely outperforms the other solution construction heuristics. On average it gives solutions within 1.74% of the in an average of less than 30 seconds of CPU. The second best result is given by the CI heuristic which produced an average percentage increase over the of 5.4% but in an average time of 0.23 CPU seconds. The relatively good performance of GI2 can be attributed in part to the local optimization procedure we apply during the insertion phase. However, this local optimization procedure requires a relatively large amount of time which explains why GI2 consumes in average 30 CPU seconds while the other construction heuristics requires in average between 0.23 and 6.65 CPU seconds.

14 14 Table 3 compares the results obtained after applying the G-opt heuristic to the solutions given in the previous table. Again the proposed heuristic produces better results. Furthermore, it is now the one requiring the smallest computation times. Notice that times for NN, CI and MO increase more rapidly than the times of GI2 when a G-opt postoptimization procedure is added. This is because the solutions produced by these heuristics are much farther from the and consequently require a larger number of improvement iterations. Table 4 compares GI3 to the other three composite heuristics (including the G-opt and G2-opt improvement phase) and shows that the GI3 produces the best results with the shortest computational times. On average it gives solutions within 0.98% of the in an average of about 83 seconds of CPU. Although the solutions produced by the MO heuristic are almost as good as those produced by the GI3, the average computation time of the MO heuristic is CPU seconds (about 47 times the computational time of GI3). Tables 5 gives a summary of the performance measures in function of problem size. It shows that the average performance of all the heuristics tested deteriorates when problem size increases. Finally Table 6 summarizes the results given in the previous tables and shows that GI3 produced the smallest average percentage increase above the and the smallest maximum percentage increase above the while requiring the smallest average computational time. This table shows that the MO or the CI heuristics followed by the G-opt improvement heuristics also produced good results. However, it seems that the MO heuristic can not be considered as an acceptable alternative as it requires much more computational time. We also applied the G3-opt heuristic (presented in section 3) to the solutions obtained by GI3 in order to see if it can further improve these results. Actually, the G3-opt heuristic improved the solution for 4 of the 36 problems: TS 225, GIL 262, PR 299 and PCB 442; giving respectively the following tour s: 68518, 1058, and Consequently, the average percentage increase above the is reduced from 0.98% to 0.74%. Unfortunately, the application of the G3-opt requires on average 1790 extra CPU seconds per problem.

15 15 6- SUMMARY AND CONCLUSIONS This paper described the GI3 heuristic, a new and efficient composite heuristic to solve the symmetric generalized traveling salesman problem. This heuristic is composed of three phases. The objective of the first phase is to construct an initial partial tour which passes through some of the node sets and visits only one node in each visited set. The second phase aims to insert one node from each non-visited set. A chain reoptimization procedure is used after each insertion to improve the obtained partial solution. The last phase is composed of two solution improvement procedures which we apply to the solution obtained at the end of the second phase. To asses the performance of GI3 we compared its solution to three other composite heuristics as well as to the optimal solution provided by Fischetti et al (1994) for 36 literature problems. GI3 produced the for 20 (55.55%) of these 36 problems. It produced solution values on the average less than 1% over the values while requiring a very reasonable amount of computation time (83.09 CPU seconds on average). However in some cases we may need a faster solution procedure. In such cases, we may use the CI heuristic (alone) which required on average 0.23 CPU seconds but produced an average percentage increase over the of 5.4%. We may also use the GI2 heuristic which produced solutions within 1.74% of the in an average of 29.5 seconds of CPU.

16 16 Table 2: Comparison of NN, CI, MO and GI 2 NN CI MO GI 2 Problem Optimal Solution EIL ST EIL PR RAT KRO A KRO B KRO C KRO D KRO E RD EIL LIN PR PR BIER PR PR KRO A KRO B PR U RAT D KRO A KRO B TS PR GIL PR PR LIN RD FL PR PCB Boldface figures indicate solutions

17 Table 3: Comparison of NN, CI, MO and GI 2 after applying the G-opt heuristic NN CI MO GI 2 Problem Optimal Solution EIL ST EIL PR RAT KRO A KRO B KRO C KRO D KRO E RD EIL LIN PR PR BIER PR PR KRO A KRO B PR U RAT D KRO A KRO B TS PR GIL PR PR LIN RD FL PR PCB Boldface figures indicate solutions 17

18 Table 4: Comparison of NN, CI and MO after applying both the G-opt and G2-opt heuristics and GI 3 NN CI MO GI 3 Problem Optimal Solution EIL ST EIL PR RAT KRO A KRO B KRO C KRO D KRO E RD EIL LIN PR PR BIER PR PR KRO A KRO B PR U RAT D KRO A KRO B TS PR GIL PR PR LIN RD FL PR PCB Boldface figures indicate solutions 18

19 19 Table 5: Performance measures in fonction of problem size Problem size Number NN CI MO GI 2 of problems Measure alone after G-opt after G-opt + G2-opt alone after G-opt after G-opt + G2-opt alone after G-opt after G-opt+ G2-opt alone after G-opt up to Average % increase above the Number of solutions nodes Average computational time between Average % increase above the & Number of solutions nodes Average computational time between Average % increase above the & Number of solutions nodes Average computational time more than Average % increase above the Number of solutions nodes Average computational time after G-opt + G2-opt Table 6: Summary of performance measures for all problems NN CI MO GI 2 Measure alone after G-opt after G-opt + G2-opt alone after G-opt after G-opt + G2-opt alone after G-opt after G-opt + G2-opt alone after G-opt after G-opt + G2-opt Number of solutions Average % increase above the Min. % increase above the Max. % increase above the Average computational time Minimum computational time Maximum computational time

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