Improving Iterated Local Search Solution for the Linear Ordering Problem with Cumulative Costs (LOPCC)

Transcription

1 Improving Iterated Local Search Solution for the Linear Ordering Problem with Cumulative Costs (LOPCC) David Terán Villanueva 1,, Héctor Joaquín Fraire Huacuja 2, Abraham Duarte 3,RodolfoPazosR. 2, Juan Martín Carpio Valadez 1, and Héctor José Puga Soberanes 1 1 Instituto Tecnológico de León (ITL), Avenida Tecnológico S/N Fracc. Industrial Julián de Obregón C.P , León, Gto. Mexico david_teran01@yahoo.com.mx, jmcarpio61@hotmail.com, pugahector@yahoo.com 2 Instituto Tecnológico de Ciudad Madero (ITCM), Av. 1o. de Mayo esq. Sor Juana Inés de la Cruz s/n Col. Los Mangos C.P.89440, Cd. Madero Tamaulipas, Mexico hfraire@prodigy.net.mx, r_pazos_r@yahoo.com.mx 3 Departamento de Ciencias de la Computación, Universidad Rey Juan Carlos, Madrid, Spain Abraham.Duarte@urjc.es Abstract. In this paper the linear ordering problem with cumulative costs is approached. The best known algorithm solution for the problem is the tabu search proposed by Duarte. In this work an experimental study was performed to evaluate the intensification and diversification balance between these phases. The results show that the heuristic construction phase has a major impact on the tabu search algorithm performance, which tends to diminish with large instances. Then to evaluate the diversification potential of the heuristic construction, two iterated local search algorithms were developed. Experimental evidence shows that the distribution of the heuristic construction proposed as diversification mechanism is more adequate for solving large instances. The diversification potential of the heuristic construction method was confirmed, because with this approach we found 26 best known solutions, not found by the tabu search algorithm. 1 Introduction In wireless communication systems it is required that cell phones communicate constantly with some base station. In order for the base station to distinguish between different signals from the cell phones, the Universal Mobile Telecommunication Standard adopted the Code Division Multiple Access technique. In this technique each cell phone has a different code. Due to the induced distortion of Authors thank the support received from the Consejo Nacional de Ciencia y Tecnología (CONACYT) and Consejo Tamulipeco de Ciencia y Tecnología (COTA- CYT). R. Setchi et al. (Eds.): KES 2010, Part II, LNAI 6277, pp , c Springer-Verlag Berlin Heidelberg 2010

2 184 D.T. Villanueva et al. radio waves propagation, each cell phone interferes with each other, so in order to preserve the connectivity it is necessary to keep the interference below an acceptable level. To reach this goal each cell phone emits a signal, and they are detected by the base station in a sequential order previously established. When a signal is detected, it is eliminated, reducing the interference to other cell phones. Then one faces the problem of finding the order of detection of cell phones to assure a proper reception for all the users. This problem is called Joint Power Control and Receiver Optimization (JOPCO) and was formulated by Benvenuto [1]. Using this problem Bertacco formulates the Linear Ordering Problem With Cumulative Costs (LOPCC) [2]. This problem is defined as follows: Given a complete digraph G=(V, A) with nonnegative node weight d i and nonnegative arcs cost c ij, the objective of LOPCC is to find a Hamiltonian path P = {p 1,p 2,..., p n 1,p n } and the corresponding node values α i that minimize the expression: where: α pi = d pi + LOP CC(P )= 1 i=n α pi n (c pip j α pj ) for i = n, n 1,n 2,...1 j=i+1 2 Related Work 2.1 Linear Ordering Problem (LOP) The linear ordering problem (LOP) is closely related to LOPCC. In this section we review some recent works related to LOP. In [4] a tabu search metaheuristic algorithm for the linear ordering problem is proposed. In the algorithm, an insertion neighborhood strategy is used along with a local search that stops once the best improvement has been reached. Laguna implements intensification and diversification besides the strategies inherent to tabu search. Additional intensification is produced with a path-relinking strategy that is applied to a set of elite solutions and with a long term memory to insert certain nodes into a complementary position related to the previously selected positions. In [5] an efficient method to perform insertion movements by using consecutive swap movements is proposed. Also this method is used in two metaheuristic algorithms to solve LOP: an Iterated Local Search (ILS) and a Memethic Algorithm (MA). In both algorithms the local search stops somewhere between first and best improvements. It is reported that the Memetic Algorithm has better performance, in time and deviation from the optimum, than the Iterated Local Search (ILS). This results were validated using a Wilcoxon non-parametric test with α =0.01. However, not enough evidence that MA has better performance than ILS for instances of size 250 is presented. A relevant contribution of this work is the economical method used to recalculate the objective function value.

3 Improving Iterated Local Search Solution for the Linear Ordering Problem Linear Ordering Problem with Cumulative Costs (LOPCC) Now we review the works directly related to the Linear Ordering Problem with Cumulative Costs (LOPCC). In [2] for the first time a formulation of the Linear Ordering Problem with Cumulative Costs is proposed. Also two exact algorithms for solving LOPCC are described. One of them is based on Mixed Integer-Linear programming, while the other is an enumerative ad-hoc algorithm that produces the best results. In [3] a tabu search algorithm for solving the LOPCC is proposed. The tabu search algorithm has four phases: construction, intensification, local search and diversification. In the construction phase a heuristic is used to generate a prespecified number of solutions, from which the one with the best objective function value is selected as the initial solution. An iteration of the intensification phase begins by randomly selecting a node. Thescorevaluefornodei in the context of the LOPCC is calculated using Eq. (1). In the intensification phase, the low score nodes have priority for selecting positions that will decrease the objective function value of the current solution. The selected node is inserted in the first position that improves the objective value, if one exists, or alternatively in the best non-improving position. Note that this rule may result in the selection of a non-improving move. The move is executed even when the move is not beneficial, resulting in a deterioration of the current objective function value. The inserted node becomes tabu-active for a pre-specified number of iterations, and it cannot be selected for insertions during this number of iterations. The intensification phase terminates after a pre-specified number of consecutive iterations without improvement. score(i) = j c ij d j + j c ji d i. (1) The local search method is used to move the best solution found in the intensification phase to a local optimum. In this process each node of the current solution is inserted in the best improving position. The diversification phase is performed for a pre-specified number of iterations. A node is randomly selected iteratively, where the probability of selecting a node is inversely proportional to its frequency count (which records the number of times that it has been chosen for moving in the intensification phase). The chosen node is placed in the best position, as determined by the move values, allowing for non-improving moves to be executed. Once a solution has been chosen as an initial solution, the global iterations of tabu search are performed. In a global iteration the intensification and diversification phases are performed. Each phase begins from the current solution andafterterminationtheyreturntheoverall best solution and a new current solution. The search terminates after a pre-specified number of global iterations have ocurred without improving the overall best solution. The reported experiments were carried out with 25 UMTS instances of each scenario, where the scenarios are synchronous and asynchronous instances with

4 186 D.T. Villanueva et al. and without scrambling. Also there were 49 LOLIB instances and 75 Random instances with a uniform distribution between 0 and 100 of different sizes, 25 instances of size 35, 25 of size 100, and 25 of size 150. A comparative study was carried out against the enumerative method proposed by Bertacco [2], a Random Greedy method proposed by Benvenuto [1] and an adaptation of the Tabu search proposed by Laguna [4]. In this work we propose to evaluate the intensification and diversification balance between the four phases of the tabu search algorithm, to define new intensification and diversification general purpose strategies using the best methods. 3 Experimental Study The goal of this experimental study is to identify the best methods used in the tabu search algorithm, in order to define new general purpose strategies of intensification and diversification. First the cost/benefit relation for the construction, intensification, local search and diversification methods of the tabu search algorithm were determined. Then the identified best methods were evaluated as intensification or diversification strategies, in the iterated local search framework. In the experimental study the algorithms were implemented in C and the experiments were performed on a computer with a Xeon 3.06 GHz processor and 4 GB of RAM. The instances used in the experiments were the same used in [3]. As the tabu search algorithm uses a predefined seed for random number generation and is time limited to 60 seconds, this parameters were used too in the experiments with the iterated local search algorithms. 3.1 Tabu Search Algorithm Diagnosis To determine the cost/benefit relation for the construction, intensification, local search and diversification methods of the tabu search algorithm, the improvement contribution and the time spent for each method were measured. Table 1 shows the results from this experiment. The first column indicates the instances set used, the second one contains the indicator used, the next four columns contain the results for the construction, intensification, local search and diversification methods. The indicators used were the improvement percentage, the percentage of the overall time and the time in seconds used by each method. As we can see, for the first five instances set the heuristic construction produces over 90% of the attained improvement with less than 20% of the time spent. For the next two sets, the construction produces over 60% of the improvement. The results for the last set shows a different pattern due to the 60-second-time restriction. In general the results show that the heuristic construction is the most important method in the tabu search algorithm, but its performance tends to diminish with large instances. As we can see the main purpose of this method in the algorithm, is to contribute to the intensification process generating a high quality initial solution. Now we consider the next question: Can we use the heuristic construction to define an efficient diversification method based on restarting the current search at different high quality solutions?

5 Improving Iterated Local Search Solution for the Linear Ordering Problem 187 Table 1. Tabu Search Algorithm Evaluation Instances Analysis Heuristic. Const. Local Search Intensification Diversification Improve % UMTS 00 Time % Time sec Improve % UMTS 01 Time % Time sec Improve % UMTS 10 Time % Time sec Improve % UMTS 11 Time % Time sec Improve % LOLIB Time % Time sec Improve % Rnd 35 Time % Time sec Improve % Rnd 100 Time % Time sec Improve % Rnd 150 Time % Time sec Diversification Based on the Initial Solution Construction Heuristic To evaluate the diversification potential of the heuristic construction, two iterated local search algorithms were developed. In this framework a general purpose diversification strategy based on initial solution construction methods is used. The initial solution construction is performed by the greedy algorithm Greedy- HeuristicConstruction(10). It is used to generate 10 solutions which are built in inverse order. For each one, the last node of the permutation is randomly selected. The next nodes are selected using a roulette that gives priority to those nodes that produce the lowest cost if they are placed at the next position of the permutation. This requires to determine the cost of the objective function for each node that could be in the next position of the current node. This is an expensive process, but it is oriented to produce very good quality solutions. Once a node is inserted, the algorithm also verifies if that node could produce a better solution if its inserted in a previous position.

6 188 D.T. Villanueva et al. The solution perturbation phase is performed by the algorithm Perturbation(p). Here a random node in the current permutation p is selected and inserted in a new position that improves the current solution, otherwise the best non-improving position is selected. The local search method is performed by the algorithm LocalSearch(p). This method inserts each node of the current permutation p in the position that produces a better objective function value. Given a node, the position where it must be inserted is determined checking the neighborhood from node position i to the left and continuing from position i to the right, using consecutive swap movements. p = GreedyHeuristicConstruction(10) p = LocalSearch(p) P better = p While globaliterationsw ithoutimprove < maxiterations and time < timelimit { ILSB: p = Perturbation(p) ILSP: p = GreedyHeuristicConstruction(10) p = LocalSearch(p) if (p < P better ) { P better = p } else { if (p > P better * β) { p = P better } } } Where: β is a numerical value between 1 and 2. Fig. 1. Structure of the ILSB(ILSP) algorithm To evaluate the diversification potential of the heuristic construction, two iterated local search algorithms were developed. The first one is a basic iterated local search (ILSB) shown in figure 1. In this algorithm the heuristic construction is used to generate a high quality initial solution, which is locally optimized, and the perturbation process is started. The solution perturbation has two goals: diversify the search because a node is randomly selected, and intensify the search because the selected node is inserted in a "good" position. In the second algorithm the heuristic construction is also used as the perturbation process. This algorithm is named ILSP and it is shown in figure 1. As we can see the heuristic

7 Improving Iterated Local Search Solution for the Linear Ordering Problem 189 construction work is distributed across all the search process. Now the perturbation continues diversifying the search, but restarting from high quality solutions. This is a diversification strategy that can be applied in several metaheuristics. For both algorithms the local search method is used to move the current solution to a local optimal solution and it is applied after the initial solution construction and the solution perturbation. The stop criterion applied has two elements: stagnation detection and a 60-second-time limit. Table 2 shows the results obtained with the UMTS instances for which the optimal values are known. These results show that the ILSP algorithm slightly outperforms the TS_Duarte algorithm in both quality and efficiency. Table UMTS instances of size 16 (25 of each group) Algorithm Object. Func. Avg. Err. Std. Dev. Num. Optima CPU sec. Synchronous TS_Duarte without ILSB scramble ILSP Synchronous TS_Duarte with ILSB scramble ILSP Asynchronous TS_Duarte without ILSB scramble ILSP Asynchronous TS_Duarte with ILSB scramble ILSP Table 3 shows the experimental results with the LOLIB instances for which no optimal solutions are known. As we can see the ILSP algorithm clearly outperforms the TS_Duarte algorithm in quality solution and efficiency. And for 16 instances, the ILSP algorithm improves the best known solutions. Table 4 shows the results obtained with the Random instances for 35, 100 and 150 nodes. For these instances the optimal solutions are unknown, and we have found 24 new best known solutions, 16 with ILSB and 10 with ILSP (2 overlapped). As we can see the TS_Duarte algorithm clearly outperforms the ILS algorithms. But as we have previously seen, the performance of the TS_Duarte algorithm is highly dependent on the heuristic construction and this process is too expensive. As we can see on the instances of size 100 and 150, the number of best known solutions obtained by the TS_Duarte algorithm decreases as the instance size increases. In contrast the number of the best known solutions obtained increases for the ILS algorithms. Also we can observe on the instances of size 150, the ILSP algorithm clearly outperforms in efficiency the TS_Duarte algorithm. The experimental evidence seems to show that the distribution of the heuristic construction used in the ILSB algorithm is more appropriate for solving large instances that the approach used in the TS_Duarte algorithm.

8 190 D.T. Villanueva et al. Table LOLIB instances of sizes 44(28), 50(2), 56(11) y 60(3) Algorithm Object. Func. Avg. Err. Std. Dev. #Best Known CPU sec. TS_Duarte ILSB ILSP Table Random instances (25 of each group) Size Algorithm Object. Func. Avg. Err. Std. Dev. #Best Known CPU sec. TS_Duarte ILSB ILSP TS_Duarte ILSB ILSP TS_Duarte ILSB ILSP Table 5. The best known solutions for the LOLIB instances Instance Value Instance Value Instance Value be75eec 5.09 t65n11xx 2.09 t75i11xx be75np t65w11xx t75k11xx 1.32 be75oi 2.79 t69r11xx t75n11xx 9.9 be75tot t70b11xx t75u11xxa stabu t70d11xx tiw56n stabu t70d11xxb 4.44 tiw56n stabu t70f11xx 1.27 tiw56n t59b11xx t70i11xx tiw56n t59d11xx t70k11xx 0.49 tiw56n t59f11xx t70l11xx tiw56n t59i11xx t70n11xx 0.05 tiw56r t59n11xx t70u11xx tiw56r t65b11xx t70w11xx 0.04 tiw56r t65d11xx t70x11xx 0.23 tiw56r t65f11xx 1.25 t74d11xx 4.76 tiw56r t65i11xx t75d11xx 5.06 t65l11xx t75e11xx

9 Improving Iterated Local Search Solution for the Linear Ordering Problem Conclusions In this ongoing investigation, the linear ordering problem with cumulative costs is approached. The best known solution algorithm for the problem is the tabu search proposed in [3].The algorithm has four phases: construction, intensification, local search and diversification. In this work an experimental study was conducted to evaluate the intensification and diversification balance between these phases. The results show that the heuristic construction has a major impact on the tabu search algorithm performance, which tends to diminish with large instances. Then to evaluate the diversification potential of the heuristic construction, two iterated local search algorithms were developed. In this framework a general purpose diversification strategy based on initial solution construction methods is shown. The experimental evidence shows that the distribution of the heuristic construction used as diversification mechanism in the ILSP algorithm is more adequate for solving large instances than the approach used in the tabu search algorithm. The diversification potential of the heuristic construction method was confirmed because the Table 6. The best known solutions for the Random instances Instance Value Instance Value Instance Value t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d t1d

10 192 D.T. Villanueva et al. ILSP algorithm found 26 best known solutions, not found by the tabu search algorithm. In tables 5 and 6 we show an update to the best known solutions reported in [3], the new best known solutions found in this work are shown in boldface. Now we are working on developing new strategies to improve the performance of the metaheuristics used to solve the large instances reported in the literature. References 1. Benvenuto, N., Carnevale, G., Tomasin, S.: Optimum power control and ordering in sic receivers for uplink cdma systems. In: IEEE-ICC 2005 (2005) 2. Bertacco, L., Brunetta, L., Fischetti, M.: The linear ordering problem with cumulative costs. Eur. J. Oper. Res. 189(3), (2008) 3. Duarte, A., Laguna, M., Martí, R.: Tabu search for the linear ordering problem with cumulative costs. Springer Science+Business Media, Heidelberg (2009) 4. Laguna, M., Martí, R., Campos, V.: Intensification and diversification with elite tabu search solutions for the linear ordering problem (1998) 5. Schiavinotto, T., Stützle, T.: The linear ordering problem: Instances, search space analysis and algorithms. Journal of Mathematical Modelling and Algorithms 3(4), (2005)