A heuristic approach for the Travelling Purchaser Problem

Transcription

1 European Journal of Operational Research 162 (2005) A heuristic approach for the Travelling Purchaser Problem Jorge Riera-Ledesma *, Juan Jose Salazar-Gonzalez DEIOC, Facultad de Matematicas, Universidad de La Laguna, La Laguna, Tenerife, Spain Received 1 November 2001; accepted 14 October 2003 Available online 24 January 2004 Abstract The Travelling Purchaser Problem (TPP) is a known generalization of the Travelling Salesman Problem, and is defined as follows. Let us consider a set of products and a set of markets. Each market is provided with a limited amount of each product at a known price. The TPP consists in selecting a subset of markets such that a given demand of each product can be purchased, minimizing the routing cost and the purchasing cost. This problem arises in several applications, mainly in routing and scheduling contexts, and it is NP-hard in the strong sense. A new heuristic approach based on a local-search scheme, exploring a new neighbourhood structure, is proposed. The key idea is to perform a k-exchange of markets instead of the classical 1-exchanges. A neighbour of a given TPP solution is another TPP solution obtained by removing a path of consecutive markets, and by inserting other markets so as to restore the feasibility. This proposal is evaluated on a broad class of instances from literature, where the routing costs are Euclidean distances. Computational results show that our proposal is favourably compared to previous heuristic algorithms in literature. Ó 2003 Elsevier B.V. All rights reserved. Keywords: Travelling Salesman; Heuristics 1. Introduction This article is concerned with a generalization of the well-known Travelling Salesman Problem (TSP), known as the Travelling Purchaser Problem (TPP). The problem can be defined as follows. Let us consider a set of n products to be purchased and a vehicle originally situated at a domicile (location v 0 ). There is a given required demand for each different product. Let us also consider a set of m * Corresponding author. Tel.: ; fax: addresses: jriera@ull.es (J. Riera-Ledesma), jjsalaza@ull.es (J.J. Salazar-Gonzalez). markets, each one selling some units (offers) of a certain number of products. The unit price of a product depends on the market. The travel cost between each two locations (domicile and markets) is also known. The TPP consists in selecting a subset of markets and routing the selected markets with a vehicle such that the demand of each product is satisfied and the total purchasing and travel cost is minimized. The same combinatorial problem can be found in a scheduling context (see Burstall [3] and Buzacott and Dutta [4]). Indeed, let us consider a set of n jobs to be processed, and a multi-purpose machine, i.e., a machine that is able to perform m different configurations. Each job requires a given /$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi: /j.ejor

2 J. Riera-Ledesma, J.J. Salazar-Gonzalez / European Journal of Operational Research 162 (2005) dedication (i.e. a set of tasks), while each configuration of the machine can execute only part of the dedication of the job. The processing time for each task of a job is given and depends on the configuration. Each task could be processed by different configurations, and each configuration could be used to perform tasks of different jobs. The time taken to change over from one configuration to another is also known. The machine remains initially in a default status (configuration v 0 ), status in which the machine must be after all the jobs are completely executed. The TPP consists in selecting and sequencing a set of configurations to fully execute the jobs minimizing the total processing and changeover time. The particular case in which there is no restricted offer of a product at each market (or where each job consists of one task) is called unrestricted TPP. It can be seen as the TPP with one-unit demand for each product. It was introduced by Burstall [3] and by Buzacott and Dutta [4]. Other applications of TPP in Job Scheduling, Warehousing and Routing Problems are described in Singh and van Oudheusden [14]. The unrestricted TPP is known to be NP-hard in the strong sense, since it becomes the TSP when each product can be purchased in only one market. Ramesh [13] described an exact method based on a lexicographical search capable of handling instances with m 6 12 and n Singh and van Oudheusden [14] presented a branch-andbound algorithm solving directed instances with m 2f10; 15; 20; 25g and n 2f10; 30; 50; 100g, and undirected instances with m 2f10; 15; 20g and n 2f15; 30; 50g. The bound in [14] is based on the Uncapacitated Facility Location Problem (UFLP), arising when the sequence requirement is relaxed. The TPP in the general case was first introduced by Laporte et al. [9]. In that article a branch-and-cut algorithm for the exact solution is also proposed and used to solve instances involving up to 200 markets and 200 products. The literature on TPP is mostly directed towards the development of heuristic or near optimal methods. One of these heuristic procedures was introduced by Golden et al. [6]. The Generalized Saving Heuristic (GSH) is proposed in the article. This algorithm starts with an initial tour containing the domicile and the market selling the largest number of products at their lowest available price. Their heuristic method was later on, modified by Ong [11] who proposed the Tour Reduction Heuristic which starts with an initial tour containing a subset of markets offering the n products and iteratively drops the markets yielding the largest cost reduction until no further improvement can be obtained. Pearn and Chien [12] suggested some improvements to the two previous works of Golden et al. [6] and Ong [11]. Two of these improvements were related to the GSH of Golden et al. [6]. Another heuristic method proposed by Pearn and Chien [12] is called Commodity Adding Heuristic. This heuristic implicitly assumes that all products are available at all markets. The first metaheuristic approaches for the TPP based on dynamic tabu search and simulated annealing are presented in Voß [15]. That article proposes two dynamic strategies for the managing of the tabu list: the Reverse Elimination Method and the Cancellation Sequence Method. Their impact on the TPP is also studied. Another metaheuristic approach is presented by Boctor et al. [2]. Several algorithms based on tabu search are introduced and tested on TPP instances. The benchmark instances have been created for both the unrestricted and the general versions, and for the particular case in which the markets are locations in the Euclidean plane. These algorithms are tested on instances up to m and n comparing the obtained heuristic solutions with their exact solutions. We propose an alternative heuristic approach for solving the TPP in the general version, i.e., with demands of products and offers at markets (or with jobs consisting of several tasks and limitations in the dedication of the configurations). The proposed heuristic is based on a local-search scheme using a special neighbourhood definition. The neighbourhood and the heuristic approach are specially designed for instances in which the routing costs are based on Euclidean distances (and particularly for the instances in [2]), but it could also be applied to solve other symmetric cost instances. The TPP is formally described in Section 2 and the proposed heuristic approach is detailed

3 144 J. Riera-Ledesma, J.J. Salazar-Gonzalez / European Journal of Operational Research 162 (2005) in Section 3. Section 4 shows several computational results on the same instances used in [2]. A comparison of the results shows our approach to be better. 2. Problem formulation The TPP can be formally described as follows. Let us consider a domicile v 0, a set of markets M :¼ fv 1 ;...; v m g and a set of products K :¼ fp 1 ;...; p n g. Let G ¼ðV ; EÞ be an undirected graph where V :¼ fv 0 g[m is the vertex set and E :¼ f½i; jš : v i ; v j 2 V ; i < jg is the edge set. For simplicity, we will assume that m P 5 and G is a complete graph. Each product p k is associated with a demand d k, and is available (i.e., with a positive offer) in a subset of markets M k M. Nevertheless, only q ki units of product p k could be acquired at each market v i 2 M k. For each p k 2 K, we assume that q ki and d k satisfy 0 < q ki 6 d k for all v i 2 M k and P v j2m k q kj P d k. Accordingly, let b ki be the price per unit of product p k at market v i. Finally, let us denote by c e the travel cost between v i and v j for each edge e ¼½i; jš 2E. Afeasible solution r of the TPP consists of a cycle in G defined by an edge subset E r E and a vertex subset V r V such that: (i) the depot is visited, i.e., v 0 2 V r, (ii) for each vertex v 2 V r the degree of v is exactly 2, (iii) it is possible to purchase the required demand, i.e. X q ki P d k ; for all p k 2 K: v i2v r \M k The set of all feasible solutions will be denoted by X. In addition, let us define the routing cost of r as travelðrþ :¼ X c e ; e2e r the purchasing cost of product p k in r as ( X X priceðr; kþ :¼ min z ki b ki : v i2v r \M k v i2v r \M k z ki z ki 6 q ki ; for all v i 2 V r \ M k ); ¼ d k ; and, the total purchasing cost of r as priceðrþ :¼ X p k 2K priceðr; kþ: Variable z ki in the definition of priceðr; kþ represents the amount of product p k to be purchased in market v i when considering the solution r. The value f ðrþ :¼ travelðrþþpriceðrþ is called the total cost of the feasible solution r represented by the cycle ðv r ; E r Þ in G. The TPP searches for a feasible solution with minimum total cost, i.e., minff ðrþ : r 2 Xg: Very special instances of the TPP arise when d k ¼ 1andq ki ¼ 1 for each p k 2 K and v i 2 M k, leading to the previously introduced unrestricted TPP. As mentioned in Section 1, most of the articles in literature are concerned with this unrestricted version. Clearly, a feasible solution r of the unrestricted TPP is a simple cycle ðv r ; E r Þ in G such that: i(i) v 0 2 V r, (ii) V r \ M k 6¼;, for all products p k 2 K, and the purchasing cost of a product p k in r is simply stated as priceðr; kþ :¼ min v i2v r \M k b ki : As mentioned Section 1, the TPP is NP-hard in the strong sense since it reduces to the TSP when m ¼ n and jm k j¼1 for all p k. The TPP also reduces to the UFLP when M k ¼ M for all p k, q ki ¼ d k for all v i 2 M and all p k 2 K, and c e ¼ðf i þ f j Þ= 2 for all e ¼½i; jš 2E, withf i the cost of opening facility v i (f 0 :¼ 0) and b ki the cost of serving customer p k from facility v i. 3. Heuristic approach This section establishes the main idea of our local-search proposal, which is based on two families of neighbourhoods. A specific procedure to achieve a local minimum is developed for each of them. The first procedure performs an iterative scheme exchanging l consecutive vertices in a given

4 J. Riera-Ledesma, J.J. Salazar-Gonzalez / European Journal of Operational Research 162 (2005) feasible cycle with a set of vertices not belonging to that cycle. The value l is reduced as soon as a local optimum is achieved. The above mentioned procedure is called l-consecutiveexchange. The second procedure inserts as many vertices as possible, whenever each insertion implies a reduction in the objective value. This procedure is called Insertion. We next describe more details on each one, starting with a data structure to speed up the evaluation of an insertion/deletion of a market in a partial solution Data structure Whenever a solution is evaluated and an insertion/deletion occurs, the evaluation of a modified solution can be efficiently recomputed using an ad hoc data structure. A partial solution r has an internal representation consisting of a sequence of markets in V r and a dynamic array for each product p k 2 K. Each component of these arrays corresponds to a market v i 2 V r \ M k and contains the offer q ki as well as the unit price b ki. These items are pre-sorted according to the purchasing cost b ki. Fig. 1(a) illustrates the array for a product p k,and Fig. 1(b) and (c) help with the idea of inserting a new market v i selling p k. According to this data structure, the insertion of a new market v i 2 M n V r in a partial solution r would take a time complexity of Oðlog jv r \ M k jþ for each product p k 2 K. Hence, the evaluation of the total purchasing cost would take a time complexity of OðjV r jjkjþ. In the particular case of the unrestricted TPP, this complexity would be reduced to Oð1Þ for each product p k 2 K l-consecutiveexchange Making use of the above data structure, the algorithm l-consecutiveexchange proceeds by exchanging a set of l consecutive vertices belonging to a feasible cycle r, with other vertices outside the cycle, in a two stage-procedure. The first stage (called l-consecutivedrop) tries to reduce the length of the cycle by removing l consecutive vertices. The second stage (called RestoreFeasibility) tries to restore the feasibility if it is lost in the previous stage. This idea generalizes the procedure IMP1 described in Voß [15], in which exactly one single vertex is removed from a feasible cycle, and a number of consecutive insertions are performed as long as an improvement in the objective function is achieved. A similar idea has also been proposed by Keller [8] for the Orienteering Problem, where two consecutive vertices are replaced by others if it leads to a better feasible route. This set of moves defines a neighbourhood with a considerable number of neighbours. Because of this, in order to select a good neighbour, the classical complete enumeration of the neighbourhood is avoided, and the following heuristic Fig. 1. (a) List for product p k previously sorted such that b kl 6 b km 6 6 b kj ; (b) and (c) evaluation of a potential insertion of product p k available at market v i, where b kl 6 b ki < b km.

5 146 J. Riera-Ledesma, J.J. Salazar-Gonzalez / European Journal of Operational Research 162 (2005) procedure is performed. Given an initial solution r, a starting value l is chosen according to a selftuning procedure. An iterative mechanism to remove each sequence of l consecutive markets is performed by the procedure l-consecutive- Drop. Whenever a modified cycle turns out to be unfeasible, the procedure RestoreFeasibility is called upon. The solution r is updated if it improves the previous one. However, if no improvement is achieved or restoring feasibility fails, value l is decreased by one unit. This procedure continues iteratively, stopping when l ¼ 0. See Fig. 2 for a pseudocode of the described procedure l-consecutivedrop This routine selects l consecutive vertices according to an estimation of the objective function reduction (i.e., the reduction in travel cost after removal and the increase in purchasing cost). For each path P E r consisting of l þ 1 consecutive edges f½s; u 1 Š; ½u 1 ; u 2 Š;...; ½u l 1 ; u l Š; ½u l ; tšg belonging to a feasible cycle r, let V ðpþ :¼ fu 1 ;...; u l g be internal vertices of P. The potential reduction in the travel cost after removing the vertices in V ðpþ is computed as follows: TravelReductionðPÞ :¼ X c e c ½s;tŠ : e2p In addition, the potential increase in the purchasing cost (and referred to as PriceIncrease(P)), as well as the set of non-satisfied products after the removal of vertices in V ðpþ, are also computed. More precisely, those units of product that were purchased in markets in V ðpþ Fig. 2. Procedure l-consecutiveexchange. have to be acquired at markets of V r n V ðpþ, adding the extra cost to PriceIncrease(P); if those units of product cannot be purchased in V r n V ðpþ then we do not penalize PriceIncrease(P) so as not to discourage the selection of a path leading to an unfeasible solution. These evaluations are performed on each possible path P with l þ 1 edges, and all of them are ranked according to TravelReduction(P) ) PriceIncrease(P). The path with the biggest rank is selected to be removed. After removing those selected l consecutive vertices, an improvement procedure is applied to reduce the routing cost of the new (and possibly non-feasible) cycle r. In our implementation this improvement is a specific version of the Lin and Kernighan [10] algorithm, available in Applegate et al. [1]. The Lin Kernighan algorithm performs a sequence of 3-opt edge interchanges, each one followed by a sequence of 2-opt edge interchanges Restoring feasibility This procedure tries to extend an unfeasible cycle r so as to restore the feasibility. To this end, new markets must be inserted. Denoting by V the set of vertices in the previous feasible cycle, the new markets are allowed to be selected from M n V to guarantee the generation of different cycles. In our experiments this decision proved to give better results than using M n V r as candidate markets. The method proceeds by computing the nonsatisfied amount d k :¼ maxf0; d k P v j2v r \M k q kj g for each product p k 2 K, and selecting a subset T M k n V of markets selling the required amount for each product. A basic greedy approach finds an initial subset T such as P v q i2t ki P d k, for all p k 2 K, by choosing the cheapest markets provider in M k n V for each product p k. However, this procedure admits the following improvement. Given a non-feasible solution r, for each vertex v i 2 M n V not belonging to r, two weights are computed. These weights are the routing increase, denoted by qðv i ; rþ, and the purchasing cost reduction, denoted by lðv i ; rþ. After computing these estimations, a subset T M n V of markets is suitably selected by solving the following problem:

6 ( X qðv i ; rþ lðv i ; rþ : X q ki P d k ; v i2t v i2t ) for all p k 2 K : min T MnV This combinatorial optimization problem is a generalization of the set covering problem, which is known to be NP-hard (see Karp [7]). Despite this, in our computational experience this combinatorial problem turns out to be easy to solve since it is concerned with small-size instances. The two above mentioned weights try to establish a discernment for the selection of the set T according to the two criteria involved in the objective function. The routing increase qðv; rþ of a market v in the current solution r, which is related to the routing cost, describes how much the routing cost could increase after the insertion of v in r. In order to calculate q, the classical saving criterion (see Clarke and Wright [5]) is used. It should be noted that, in contrast to the case in which the saving is related to a single vertex, there are some cases in which the sum of the individual saving costs is not an upper bound of the increase in the travel cost. More precisely, let us denote the sum of the individual routing cost of a vertex set T M n V by qðt ; rþ :¼ X qðv; rþ; v2t then condition J. Riera-Ledesma, J.J. Salazar-Gonzalez / European Journal of Operational Research 162 (2005) qðt ; rþ P travelðr 0 Þ travelðrþ; ð1þ where r 0 is the feasible cycle obtained after inserting the vertex set T in r, does not hold in some cases. Indeed, let us focus on a subset T to be inserted between two vertices s and t belonging to the cycle r, and let P be a path through the vertices of T with extreme vertices s and t. According to the classical saving criterion, each single insertion of a vertex v 2 M n V is qðv; rþ :¼ c ½s;vŠ þ c ½v;tŠ c ½s;tŠ : Hence, we have that travelðr 0 Þ travelðrþ ¼ X e2p ¼ c ½s;u1 Š þ c ½ul ;tš c ½s;tŠ þ Xl 1 c ½ui ;u iþ1 Š: In addition, i¼1 qðt ; rþ ¼c ½s;u1 Š þ c ½ul ;tš c ½s;tŠ þ Xl 1 l¼1 ðc ½s;uiþ1 Š þ c ½ui;tŠ c ½s;tŠ Þ: c e c ½s;tŠ ð2þ ð3þ Therefore, from (2) and (3) it follows that condition (1) holds when X l 1 i¼1 ðc ½s;uiþ1 Š þ c ½ui;tŠÞ P Xl 1 ðc ½ui;uiþ1 Š þ c ½s;tŠ Þ: i¼1 ð4þ Fig. 3(a) illustrates a simple example evaluating the travel cost for the insertion of a path between s and t. In this particular case the condition (4) does not hold, since c ½s;uiþ1 Š þ c ½ui;tŠ < c ½ui;uiþ1 Š þ c ½s;tŠ for i ¼ 1; 2; 3 (see Fig. 3(b)). In spite of this drawback, the criterion of estimating TravelReduction(P) by adding the single savings worked well in our experiments, as shown in the Section 4. The purchasing cost reduction lðv; rþ describes the reduction in the purchasing cost after the Fig. 3. Evaluation of the increase in the travel cost of a set of vertices to be inserted between the vertices s and t.

7 148 J. Riera-Ledesma, J.J. Salazar-Gonzalez / European Journal of Operational Research 162 (2005) insertion of a single vertex v in the current solution r, since a cheaper product may be provided by this new market. For each product p k and each vertex v i 2 M k n V, the savings in the purchasing cost price(r 0 ) of the new cycle on V r0 :¼ V r [fv i g can be efficiently computed by inserting v i in the dynamic array described in Section 3.1. More precisely, r 0 inherits from r the amount of p k purchased in markets v j 2 V r when b kj 6 b ki, while the other amount must change taking into account the new market v i. In particular, it is convenient to purchase r ki ðv r Þ 8 8 < < :¼ min q ki ; max 0; d : : k X v j2v r \M k ;b kj 6 b ki q kj 99 = = ;; units of p k in the new market v i. The same amount of units of p k must be not purchased in the markets of V r \ M k with b kj > b ki, and the computation of this adjustment is immediate using the data structure pointed P out in Section 3.1. The total decrease p k 2K ½priceðr0 ; kþ priceðr; kþš is considered as the insertion price of v i in r, and denoted by qðv i ; rþ. By using the stated dynamic data structure the theoretical complexity is equivalent to the direct evaluation of the pricing cost, but in practice we notice that the above described procedure reduces the computational effort. Finally, the procedure inserts the vertices in the selected T, one after the other, by considering the maximum saving criterion. The obtained cycle is re-optimized by using the Lin Kernighan refinement, already mentioned at the end of the l- ConsecutiveDrop section. qðv; rþ lðv; rþ > 0 is inserted in r if such a vertex exists General algorithm An initial solution containing all vertices is built by the well-known nearest-neighbour TSP heuristic, and it is then improved by the previously mentioned Lin Kernighan procedure. Afterwards, the following iterative scheme is performed. An inner loop computes the value l, as mentioned in Section 3.2, and obtains iteratively better solutions by applying l-consecutiveexchange and Insertion, until no further improvement is achieved. Once the inner loop concludes, a perturbation procedure is carried out in order to augment the current cycle with new vertices. A perturbation scheme (called Shaking) controls both the number of added vertices and the number of iterations of the outer loop. In particular, for a given solution r, each vertex not in V r enlarges r if the travel cost of the augmented solution does not increase by more than u percent. The percentage u is iteratively reduced and the procedure stops when no vertex is inserted. The choice of the values for u was taken based on our computational experiences: initially u :¼ 35 and iteratively it is reduced by one unit. See Fig. 4 for a pseudo-code illustrating the general procedure. The aim of this perturbation scheme is to provide different initial solutions to the above procedure so as to avoid local minimum solutions Insertion The procedure Insertion adds a new vertex to the current feasible cycle r if such insertion implies a reduction in the total cost of r. In order to perform this procedure both the qðv; rþ and lðv; rþ are computed for each vertex v 2 M n V r. The vertex maximizing qðv; rþ lðv; rþ such that Fig. 4. General algorithm.

8 J. Riera-Ledesma, J.J. Salazar-Gonzalez / European Journal of Operational Research 162 (2005) Computational results Our proposal has been tested on the series of randomly generated problems described in Boctor et al. [2], containing instances for the restricted and unrestricted TPP versions. More precisely, m þ 1 randomly generated points have been located in the square ½0; 1000Š½0; 1000Š according to a uniform distribution and defining routing costs by Euclidean distances. The first location corresponds to the domicile. Each product p k has been associated with jm k j randomly selected markets, where jm k j has been randomly generated in interval ½1; mš. Product prices b ki are generated in the interval ½1; 500Š according to a discrete uniform distribution. For the restricted case, limits on supplies and demands have also been generated in the following way. For each product p k and each market v i, q ki has been randomly generated in [1,15] and d k :¼dkmax vi 2M k q ki þð1 kþ P v i2m k q ki e for k ¼ 0:1, 0.3, 0.5, 0.7, 0.8, 0.9, 0.95 and Note that, the bigger the k value is, the shorter the length of its optimal cycle is. For instance, with k ¼ 0 the instance becomes a TSP, while with k ¼ 1 it becomes the unrestricted TPP. Five instances were generated for each value of n, m and k. Therefore, the first family contains 140 cases and the second family contains 960 cases. Tables 1 and 2 compare our results with [2] on the unrestricted and restricted TPP instances, respectively. Columns CAH1, CAH2, UPH1, UPH2 and CPH correspond to the different approaches proposed by Boctor et al. [2]; and columns LS correspond to the local search algorithm described in this article. Each column shows the quality of the heuristic solution over the optimum solution obtained by using the exact method described in Laporte et al. [9] (column %gap), and the CPU seconds consumed by the heuristic approach on a PC Celeron 500 MHz (column Seconds). Each row contains the average results related to the subset of instances solved to optimality by the exact method and grouped according to the value m, n and k. The column denoted by # gives the number of instances involved in each row (i.e., the number of instances with a known optimal solution from the exact method described in [9] using a time limit of 2 hours of the Celeron 500 MHz). The column %Visited shows the average number of markets involved in an optimal solution computed by the exact algorithm described in [9]. Tables 1 and 2 clearly show that our approach provides solutions very close to the optimal ones. On the restricted TPP instances (more difficult than the associated unrestricted ones), the average computational time was close to one minute of the PC Celeron 500 MHz. Solving the set covering subproblems (by calling RestoreFeasibility) took about 7% of the total computational time. Even if the set covering problem is a hard problem, the short time consumed in our experiments is explained by the small size of the instances of the subproblem we solved. Both quality and consumed time on these small/medium instances were Table 1 Average computational results, unrestricted (optimal) instances %Visited # Boctor et al. [2] CAH1 CAH2 UPH1 UPH2 LS %gap Seconds %gap Seconds %gap Seconds %gap Seconds %gap Seconds m n

9 150 J. Riera-Ledesma, J.J. Salazar-Gonzalez / European Journal of Operational Research 162 (2005) Table 2 Average computational results, restricted (optimal) instances %Visited # Boctor et al. [2] CAH1 CAH2 CPH LS %gap Seconds %gap Seconds %gap Seconds %gap Seconds m n k not so dependent on k and n as on m. The quality is slightly better when k approximates to 1 because k ¼ 0 produces TSP instances, while k ¼ 0:9 produces instances involving both the optimal routing and selection of markets. This conclusion coincides with similar studies on other routing-location problems (see, e.g., Keller [8]). Regarding both the quality of the solutions and the computational effort, we observe that our local search proposal improves on the approaches proposed in Boctor et al. [2]. The quality of our heuristic approximation for selecting the vertices to be added in the procedure RestoreFeasibility is measured in Table 3. It shows the maximum and average percentage of the difference between this heuristic and the exact choice. More precisely, procedure RestoreFeasibility, which selects a set of vertices based on estimations of the reduction in the objective function, has also been solved optimally using the branch-and-cut code described in [9]. It is observed from this table that RestoreFeasibility selects and inserts new markets with a gap of 2.5% of error. In spite of this gap the procedure proved to be effective in our experiments. In order to choose the initial value of parameter l, as well as to prove that the l-consecutive- Exchange has a better performance than the classical approach in which exactly one drop and several adding moves are performed, the following experiments have been carried out. For each benchmark restricted TPP instance with k 2f0:8; 0:9; 0:95; 0:99g, the procedure l-consecutive- Exchange has been executed with different initial values of l. Notice that this procedure starts from a TSP solution, iteratively removes sequences of consecutive vertices and inserts others to restore feasibility, as described in Section 3.2. The objec- Table 3 Average and maximum value of the percentage of the difference between the optimal and heuristic estimation in procedure RestoreFeasibility n m Average Maximum

10 J. Riera-Ledesma, J.J. Salazar-Gonzalez / European Journal of Operational Research 162 (2005) neighbour is obtained by removing a path of consecutive vertices and by inserting a new one so as to restore feasibility. The performance is favourable compared with other tabu search approaches recently proposed to solve Euclidean travel cost instances in literature. This heuristic approach can also be applied to other similar problems where minimum cost non-hamiltonian cycles must be located in a graph subject to additional constraints. Fig. 5. Deviation of the objective function and CPU time from their minimum values for different values of l. tive value of the generated TPP solutions and the computational time have been normalized in the rank ½0; 1Š with respect to the minimum and maximum values obtained varying the parameter l. The average (normalized) values are computed for each l and Fig. 5 plots them those points associated with the objective function with boxes and those points associated with the consumed time with circles. For very small values of l, the computational time is close to being proportional to the number of iterations thus, e.g., it is greater for l ¼ 1 than for l ¼ 3. Moreover, big l values imply strong modifications of the cycle, hence the gap and the computational effort increase with l. The best results are obtained for l between 2 and 25, thus inspiring our proposal for managing l in the procedure. We have also experimented with the described approach on (restricted and unrestricted) TPP instances involving up to m ¼ 350 and n ¼ 200, obtaining similar performances when comparing the quality of the heuristic solution with the LPrelaxation of the model in [9]. The average gap was close to 0.5% while the computing time was never more than one minute. 5. Conclusions We have introduced a local search heuristic for a location-routing problem known as Travelling Purchaser Problem. The basic idea is the definition of an appropriate neighbourhood explored by using a heuristic procedure. Given a solution, each Acknowledgements This work has been partially supported by Gobierno de Canarias (PI2000/116) and by Ministerio de Ciencia y Tecnologıa (TIC C06-02), Spain. We thank Jacques Renaud for providing us with the source code used by Boctor et al. [2]. References [1] D. Applegate, R. Bixby, V. Chvatal, W. Cook, Concorde: A code for solving traveling salesman problem, Available from [2] F.F. Boctor, G. Laporte, J. Renaud, Heuristics for the traveling purchaser problem, Computers & Operations Research 30 (2003) [3] R.M. Burstall, A heuristic method for a job sequencing problem, Operational Research Quarterly 17 (1966) [4] J.A. Buzacott, S.K. Dutta, Sequencing many jobs on a multipurpose facility, Naval Research Logistics Quarterly 18 (1971) [5] G. Clarke, J.W. Wright, Scheduling of vehicles from a central depot to a volume of delivery points, Operations Research 12 (1964) [6] B.L. Golden, L. Levy, R. Dahl, Two generalizations of the traveling salesman problem, Omega 9 (1981) [7] R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller, J.W. Thatcher (Eds.), Complexity of Computer Computations, Plenum Press, New York, 1972, pp [8] C.P. Keller, Algorithms to solve the orienteering problem: A comparison, European Journal of Operational Research 41 (1989) [9] G. Laporte, J. Riera-Ledesma, J.J. Salazar-Gonzalez, A branch-and-cut algorithm for the undirected traveling purchaser problem, Operations Research 51 (6) (2003)

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