# MODERN HEURISTIC TECHNIQUES FOR COMBINATORIAL PROBLEMS PDF

Faced with the challenge of solving hard optimization problems that The meta- heuristic approach called tabu search (TS) is dramatically. Meta-heuristics are the most recent development in approximate search methods for. solving complex optimization problems, that arise in. MODERN HEURISTIC. TECHNIQUES FOR. COMBINATORIAL PROBLEMS. Edited by. COLIN R REEVES BSc, MPhil. Department of Statistics and Operational.

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Modern heuristic techniques for combinatorial problems. Fred Glover. Loading Preview. Sorry, preview is currently unavailable. You can download the paper by . computation, and other heuristic optimization techniques, and how they are problems, to compare and judge the efficacy of modern heuristic optimization. Modern Heuristic Techniques for Combinatorial Problems by Colin R. Reeves, , available at Book Depository with free delivery.

Applications of Modern Heuristic Methods , Ed. Ducek and T. Scheuer Threshold accepting: Durbin and D. Eglese Simulated annealing: A Feo, M.

## Modern Heuristic Techniques for Combinatorial Problems

Resende, and S. Smith A greedy randomized adaptive search procedure for maximum independent set, Operations Research 42 , Feo and M. Feo, and M. Resende, A probabilistic heuristic for a computationally difficult set covering problem, Operations Research Letters , 8 , 67— Fox Integrating and accelerating tabu search, simulated annealing and genetic algorithms, Annals of Operations Research , 41 , 47— Garey and D.

Johnson Computers and Intractability: Freeman and Company, New York. Gendreau, A. Hertz, and G. Laporte A tabu search heuristic for the vehicle routing problem, Management Science 40 , — Ghaziri Supervision in the self-organizing feature map: Application to the vehicle routing in: Osman and J. Kelly, Kluwers Academic Publishers, Boston. Glass and C. Potts A comparison of local search methods for flow shop scheduling, Annals of Operations Research , 60 Forthcoming.

Glover, E. Pesch and I. Glover, J. Kelly, and M.

Glover, M. Laguna, E. Taillard and D.

Baltzer Science Publishers, Basel, Switzerland. Glover Tabu search: Improved solution alternatives, in: Mathematical Programming, State of the Art , Eds. Brige and K. Murty, The University of Michigan, Michigan. Glover and A.

## Modern Heuristic Techniques Combinatorial Problems

Glover Tabu thresholding: Glover Ejection chains, reference structures and alternating path methods for the travelling salesman problems, Working paper, Graduate School of Business, University of Colorado, Boulder, CO Glover Future paths for integer programming and links to artificial intelligence, Computers and Operations Research , 1 , — Hart and A.

Shogan Semi-greedy heuristics: Hasan and I. Osman Local search algorithms for the maximal planar layout problem, International Transactions in Operations Research , 2 , 89— Hopfield and D.

Tank Neural computation of decisions in optimization problems, Biological Cybernetics , 52 , — Husbscher and F. Ingber Simulated annealing — practice versus theory, Mathematical And Computer Modelling 18 , Jones Genetic algorithms and their applications to the design of neural networks, Neural Computing and Applications , 1 , 32— Laguna and F.

Soriano and M. Gendreau Diversification strategies in tabu search algorithm for the maximum clique problem, Annals of Operations Research , 60 Forthcoming. Kirkpatrick, C. Gelatt, and P.

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The algoritm is shown in Fig. However, in modified RTS algorithm, the tabu list size depends on the number of iterations when the solutions do not override the aspiration level. The judgment was made since the repetition of configurations in TSPs is less significant as the number of cities increases.

In modified RTS, an initial tabu list size was generating based on the number of available cities and keep the memory on the search process.

The algorithm then searches the best neighborhood solution as the new current solution for the next iteration and keeps updating the search process.

The tabu list size is increased by 1 if the solutions are not improved for a specified number of iterations and reset back to the initial tabu list size if it achieves a specified value of the tabu list size. Five dataset will be used to evaluate the performance of the proposed algorithms. These five chosen benchmark problems of symmetric TSPs are hagaregn20, wi29, dj38, eil51 and eil76 ranging in the size of cities from These data sets were obtained through three different open access websites.

The locations of the cities in the data sets are displayed in node coordinates. The distance between two cities is computed using the Euclidean distance Eq.

Different combinations of annealing schedules were used to select the suitable cooling rate and initial temperature. Six different cooling rates i. Each combination was solved by a number of independent runs each of which consisted of fifty trials.

For each trial, the minimum distance was recorded. The average distance results for the tour from the fifty trials are summarized in Table 1.

The data in Table 1 were run as a randomized complete block design to determine the significant difference between the cooling rate and initial temperature.

The outputs are shown in Table 2 and 3. Table 3 shows that different cooling rates can affect the performance of SA since there is a significant difference among the five different cooling rates. However, there is no significant difference in performance of SA with different initial temperatures.

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Table 1: Average distance obtained from 50 trials Fig. The parameter values for SA are summarized in Table 4. Each benchmark was solved using a number of independent runs; each of which consists of thirty trials. For each trial, a tour was determined by the four variant heuristics algorithms.

The parameter settings for the SA algorithm were determined through the empirical testing and statistical analyses from the first experiment as shown in Table 4.

Meanwhile, the parameter settings for the TS algorithm are shown in Table 5. For each trial, the minimum distance of the tour and its computational time were recorded. For each algorithm, the relative differences, RDbs and RDav, to access its performance will be computed. The RDbs and RDav are computed as follows: The relative differences were chosen as performance indexes in percentage to compare the performances of the four algorithms.

A smaller value of the index indicates better algorithm performance, where the performance index of 0 is the optimal performance. The results of the relative differences for the problems are shown in Table 6 and 7. The Opt column shows the benchmark solutions as reported in the literature. Based on the descriptive statistics in Table 6 and 7 , all the performance indexes of the modified RTS algorithm for both RDav and RDbs were relatively smaller than the performance indexes of the other algorithms.

Apart from the descriptive statistics, an inferential statistical test using IBM SPSS Statistics19 also have been employed to discover any significant differences among the performances of the three TS algorithms i. For each data set, there are two kinds of independent sample tests based on the assumptions of the data which are normality and homogeneity of variance. Table 8: Results of assumption tests If both of the assumptions are met, the parametric test i.

Otherwise, if either one assumption is violated, the nonparametric test i.

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Based on Table 8 , only the 5th TSP data set i. However, the variances among the TS algorithms were not constant for the eil76 problem through the levene test. Therefore, the assumptions of the data have been violated for all of the data sets.

Thus, Kruskal Wallis test was used in the further data analysis. The results of the Kruskal Wallis test are shown in Table 9. This study extends the work of Lim et al.

In brief, they first introduced the ITS algorithm integrating two heuristic algorithms i. To improve the solution quality, they then developed the RTS algorithm which dynamically adjusts its tabu list size. In the mean time, Hong et al.

They showed that with the right setting of an annealing schedule, a good solution quality could be achieved. These four algorithms were analyzed using theoretical analyses and empirical testing. From the analysis of the results, the best algorithm for solving symmetric TSPs can then be identified and reported. If this process is allocated with enough time, SA could then find the optimal solution of a considered problem. Based on this analogy of how metal is cool and annealed, each step of the SA algorithm replaces the current solution by a random nearby solution which is gradually decreased during the searching process.

The basic procedure of SA starts with an initial solution which then gradually moves to a nearby solution obtained by a local search. In order to obtain the nearby solution, 2-opt switch procedures which swap 2 cities have been used.

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The SA algorithm could escape from a local minimum by accepting non-improving moves based on the transition probability controlled by two factors i.

The basic algorithm of SA is as in Fig. The algorithm continues searching as temperature declines and stops once the temperature becomes zero. Thus, the efficiency of the annealing process is significantly affected by its annealing schedule which is controlled by the initial temperature, the cooling rate and the number of iterations which should be performed at each temperature.

A favorable annealing schedule through the use of a statistical framework for solving symmetric TSPs has been discussed in detail by Hong et al. It utilizes a hill climbing search strategy based on a set of elementary moves to diversify the search and a systematic use of memory to avoid any traps at local optimal points. Its key strategy is to move from solution to solution by accepting non-improving moves to the best solution in the neighborhood of the local optimal The main advantage of TS lies in the intelligent use of memory along with responsive exploration in the solution space The TS does not remember the current and the best solution.

However, it keeps memory on the tourthrough the last solution visited to guide the move from the current to the next solution. In TS, the memory ability is represented by its tabu list size. The use of memory helps intensify in elite regions or diversify the search towards unexplored regions.

The balance between the intensification and diversification strategies is used to control and run the search process. To ensure an efficient search process, TS requires search parameters whose values significantly depend on the types of problems. Parameter tuning especially for the tabu list size is often needed to obtain good results However, since its performance depends significantly on its initial solution 30 , Lim et al. In order to dynamically tune the tabu list size, they then presented the modified RTS algorithm to achieve a good balance between intensification and diversification.

A new solution is generated by a move which swaps two cities at the current solution. These search processes continue until stopping criteria are met e. An initial solution chosen from a local optimal configuration will generate a high quality solution.

Lim et al. Figure 2 shows the algorithm of ITS. At the beginning, the SA algorithm have been used to generate an initial solution for the ITS algorithm. After that, its neighborhood was explored to get the best neighborhood solution as the current solution using the 2-opt switch procedures. Next, TS algorithm was performing to search the best neighborhood solution as the new current solution for the next iteration. The search process continues until stopping criteria are met. In ITS, the memory on the search process is do not preserve and the tabu list size is always static which is set based on the number of cities of a considered problem.

Basically, RTS attempts to remedy the difficulty in choosing an appropriate tabu list size without prior knowledge about the search space structure. The RTS algorithm maintains the basic steps of TS except that its tabu list size is adaptive to a considered problem and the current solution of the search process. Through this appropriate parameter values, a good balance between intensification and diversification can be achieved without requiring a lot of prior experience or appropriate parameter values of the problem.

Thus, RTS is one the reactive search methods employing two mechanisms i.Adewole et al.

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Apart from the solution quality, the performance of the algorithms was also evaluated based on the computational time needed to obtain the solution. Shogan Semi-greedy heuristics: More options. Souvik Saha added it May 03,