САМОКОНФИГУРИРУЮЩИЙСЯ ЭВОЛЮЦИОННЫЙ АЛГОРИТМ ДЛЯ РЕШЕНИЯ ЗАДАЧИ КОММИВОЯЖЕРА

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Considered way of GA self-configuration was introduced in [1] where its usefulness was demonstrated on benchmark problems and in applied problems of neural networks weights adjustment. This approach was then successfully used in solving real world optimization problems with algorithmically given functions and mixed variables [2]. This made the approach to be a candidate for the development of adaptive algorithms of combinatorial optimization. This paper considers self-configuring GA, selfconfiguring ACO and their application in one of the most known combinatorial optimization problem named Travelling Salesman problem (TSP). The traveling salesman problem belongs to the class of NP-complete problems, is often used to test the newly created algorithms of combinatorial optimization and it has a lot of applications in routing, scheduling and many other fields [3]. Performance of these algorithms is compared with other heuristics namely Lin-Kernigan heuristic and Intelligent Water Drops algorithm. Results of numerical experiments on benchmark problems show that suggested approach demonstrates competitive effectiveness. Algorithms description Lin-Kernighan heuristic. One of the classical methods for solving the traveling salesman problem is a local search [4], in particular so called k-opt algorithm (Lin-Kernighan heuristic [5]). TSP solution is presented by cyclic graph f. The k-opt neighborhood Nk(f) includes all the tours which can be obtained by removing k edges from the original tour f and adding k different edges such that the resulting tour is feasible. The essence of the algorithm is to consider neighborhood of current solution. If there exists a graph g in this neighborhood with better objective function value, then g becomes current solution. The procedure is repeated as long as the current solution can be improved. Exploring the whole Nk(f) takes O(nk) operations and, thus, 2-opt and rarely 3-opt are used in practice. This paper deals with 3-opt because it is more efficient. Intelligent water drops algorithm. Intelligent water drops algorithm (IWDs) [6] possesses a few properties of a natural water drops. The paths that a river follows have been created by a swarm of water drops. Thus, any swarm of water drops will influence the rivers path. On the other side, for a swarm of water drops, the river is the part of the environment that has an influence over it. A large influence on the movement of the river shows which type of soil and how resistant it is to the flow, as it determines the speed drops. Thus, the path of the water drops swarm depends on path of the river, type of soil and its resistance. So the formation of a natural river is the result of a competition between the water drops and the environment that resists its movement. Notice that all natural rivers are full of twists and turns. This is due to the influence of gravity which pulls the water through the path of least resistance to the lowest point. It is assumed that each drop of water is able to transfer an amount of soil from one place to another. Furthermore, the soil is transferred from the fast parts of the river to the slow parts. This makes the fast parts deeper, allowing them to hold a greater volume of water. The quantity of soil a water drop is able to transfer depends on its velocity. Furthermore, the velocity of a water drop depends on the amount of soil in its way. The velocity of a water drop grows faster on a path with less soil. Water drops prefer a path with the least amount of soil. On the basis of the above properties Shah-Hosseini in 2007, proposed the Intelligent water drops algorithm [6]. Every intelligent water drop (IWD) has two important properties: the amount of soil that it carries, and its velocity. For each IWD, the values of both properties, soil and velocity, may change as the IWD flows in its environment. From the mathematical point of view, the environment is a problem for river sand swarm of water drops looking for the optimal path. Velocity of IWD, that moves from its location i to the location j, is increased by an amount a Avel = bv + cvsoil (i, j) where parameters av, bv and cv should be chosen as positive numbers. IWD’s soil is increased by removing some soil of the path ij. The amount of soil added to the IWD is calculated by n Asoil (i, j) = bs + cs ■ time2 (i, j; velIWD ) where as, bs and cs are positive parameters. Time is calculated by the simple laws of physics for linear motion. time(i, j; velIWD (t +1)) = j vel Local heuristic function HUD(. , .) has been defined for a given problem to measure the undesirability of an IWD to move from one location to the next. For TSP it is calculated as follow: HUD(i, j) = I e (i) - e ( j)|| where e (i ) is vector of coordinates, || - Euclidean metric. Soil amount between i and j is updated by the amount of soil removed by the IWD by formula: soil(i, j) = (1 - Pn ) ■ soil(i, j) - Pn ■ Asoil(i, j) The soil of the IWD is increased by the amount of soil as shown below: soilIWD = soilIWD +Asoil (i, j) The probability of choosing location j after i is proportional to the amount of the soil on the path between locations i and j and can be calculated by formula: f (soil (i, j)) pIwd (j)= Σ f (soil(i, k)) ktVv where and f (soil (i, j)) = 1 Ss + g (soil (i, j)) g (soil(i, j)) = soil (i, j) if min(soil(i,l)) > 0 leVc soil (i, j) - min(soil (i, l)) leVc otherwise 135 Вестник СибГАУ. № 4(50). 2013 The constant parameter ε^ is a small positive number to prevent division by zero. Vc is a list of visited nodes. Effectiveness of the algorithm depends on many parameters. Some of them (velocity updating parameters av, bv and cv, soil updating parameters as, bs и cs, global soil updating parameter ρΙψΰ) were fixed in our experiments according to the recommendations of the algorithm's author (1, 0.01, 1, 1, 0,01, 1 and 0,9 accordingly). However, the remaining parameters required settings for a specific task (significance of the best solution in upgrading of soil matrix α, local soil updating parameter ρ„, initial soil on each edge of the graph InitSoil, initial velocity of each drop (InitVelocity). Each parameter can take a large number of values, but in this study we did not set ourselves the aim of fine-tuning algorithm for a specific task and therefore considered only 24 variants: α = 0,1, 0,3 or 0,5, ρη = 0,9 or 0,7, InitSoil = 1000 or 10000 and InitVelocity = 20 or 200. Ant colony optimization algorithm. Ant colony optimization algorithm (ACO) [7] is a nature-inspired optimization meta-heuristic based on the behavior and organization of ant colonies in their search for a food. Being almost blind animals, ants anyway can find shortest path from the nest to the food. For information exchange, ants use a ferment, or more exactly pheromone, that they leave on the traversed path. The probability that the ant will choose a certain path is proportional to the amount of pheromone on it. Solutions in ACO are represented as permutation of n cities and ants chose next city using taboo-list (list of visited cities) and pheromone matrix at every stage. ACO has some adjustable parameters: evaporation rate (ρ), relatively importance of previous search experience (α) and relatively importance of the distance between cities (β). Pheromone trails are updated after each ant has completed a tour by formula: x,j(t+1) = p-Xj(t) + Aiij, where ρ is parameter such that (1 - ρ) is evaporation and Δτ, Σ Δτ k k=1 Here Δτ j, is an amount of the pheromone that the ant k leaves on the edge ij and can be calculated by formula: Δτ k = —, if ant k uses edge ij in its tour 0, otherwise Pp = i b (t ) ГЬ T Σ [(t)Γ·[η* ] keallowedk 0 , if j є allowedk if j i allowedk, where allowedk is a list of unvisited by k-th ant cities. Some parameters, such as — and ρ, does not significantly affect the efficiency of the algorithm, so in this paper we will consider only the parameters α and β. Genetic algorithm. A well-known genetic algorithm (GA) [8] is based on some principals of evolution, but in the GA for the TSP a chromosome is represented as permutation of the n numbers (number of cities). That is why some standard operations have a few changes, but many adjustable parameters remain such as mutation probability, the type of selection, etc. There are three type of selection in genetic algorithm: 1) Tournament selection (parameter is size of the tournament). First, we select a random subset of k individuals from the population and then select the best solution out of this subset. 2) Fitness proportional selection. The probability of the i-th individual to be selected is proportional to its fitness function value fiti and is calculated as follows: fiti Pi =- Σ fitJ j=1 , where m is a number of individuals. 3) Rank selection (linear or exponential ranking). Individual i has rank less than individual j (Ri < R,) if i-th fitness function value is less than j-th fitness function value (fitt < fitj). 3.1) Linear ranking. Probability of the i-th individual to be selected is calculated by formula: Ri Pi =■ Σ Rk k=1 , where Σ Pk =1 k=1 3.2) Exponential ranking. All individuals are assigned a weight according to the value of fitness function so that the best individual has weight ωι = 1, (k + 1)-th individual has weight , if Rk+1 = Rk, ®k+1 = ω, ·λ, otherwise, where λ є [0;1]. In this case probability of the i-th individual to be selected is where Q is a constant, Lk - length of the k-th ant tour. Let η j = 1/dj· (dj is distance between i and j) be called a visibility. The probability of choosing the city j after i is a function of the distance between cities and amount of pheromone on the edge ij and can be expressed as follows Pi =-· 1 i _m Σ ω j j=1 Algorithms self-configuration. All bionic algorithms have many adjustable parameters and this is a significant disadvantage because parameter tuning is a difficult task even for the specialists. Nowadays, for elimination of this defect one applies the tuning algorithm parameters during its work or, it can be said, an adaptation [9]. In this study we have investigated the adaptive GA, which had 8 different selection variants - tournament selection with size of the tournament equals 2, 4 or 8, rank selection with a linear ranking, rank selection with exponential co 136 2nd International Workshop on Mathematical Models and its Applications ranking with parameter λ equal to 0,95, 0,8 or 0,5, and fitness proportional selection. Also it had 5 different mutation variants - very low, low, medium, high and very high. Adaptive ACO had 4 different variants of parameter α and also 4 variance of parameter β, in both cases 1, 2, 5 or 10. Variant of each operator shall be determined separately. Let z be a number of variants of operator. In our situation, for example, z = 8 in case of selection operator of adaptive GA. If we take a parameter of the algorithm instead of the operator, and different values of this parameter instead of operator variants, then z = 4 in case of parameter α of ACO. So here the essence of adaptation will be described in terms of operators and their variants. At the beginning of the algorithm probability of selecting all the options of the operator are the same: p = 1/z. At each generation, the effectiveness of each operator variant is estimated in the following way: ni Σ/· averagefitnessi = j-, i = 1,2,..., z ni where ni is a number of individuals obtained by the i-th variant of the operator; fij is the value of the fitness function of the j-th individual obtained by the i-th variant of the operator; averagefitnessi is the average value of the fitness function of individuals that were generated by the i-th variant of the operator. Probability of the operator variant with the largest value of average fitness (i.e. the most effective) increases by ((z - T)■K)/(z■N), while the probability of all other operator variants decreases by K/(zN)), where N is the number of past generations of the algorithm, K is a constant, usually equals to 2. In addition, there must be a lower bound of the probability of the operator variant as no one of its probabilities can be equal to zero. If some probability reaches lower bound, this variant stops to give its share in the benefit of the best variant. The sum of probabilities of all variants of the same operator is always equal to 1. Thus, the probability distribution of the operator variant selection is gradually displaced towards the most effective operator variant from less effective variants. Thereby, when the algorithm has to create the next offspring from the current population, it firstly must configure settings, i.e. form the list of operators with the use of operator probability distributions. Then the algorithm selects parents with the chosen selection operator, produces an offspring with the chosen crossover operator, mutates this offspring with the chosen mutation probability and puts it into an intermediate population. When the intermediate population is complete, the fitness evaluation is executed and the operator rates (probabilities to be chosen) are updated according to the operator’s productivity, i. e. the ratio of the average offspring’s fitness obtained with this operator and the offspring population average fitness. Algorithms work analysis. Algorithms performance was compared on two well-known benchmark problems Oliver30 and Eil51. To solve these problems all heuristics have got as much resource as algorithm 3-opt required on average (i. e., 52800 objective function calculations in case of Oliver30 and 342210 in case of Eil51). Results of numerical experiments averaged on 100 runs are presented in table 1. Figure shows one example of algorithms work on Oliver30 problem in case where the better solution (423.741) was found. Here we can see some advantage in the effectiveness of ACO that may be due to the fact that it begins its work with a solution oriented on the distances between cities, but not with a completely random as other algorithms. Table 1 shows that adaptive methods lose conventional with the best settings not so much. At the same time, they do not require testing many variants of parameters. Effectiveness of biology inspired algorithms on a specific task varies considerably depending on the settings. If we solve the problem only once (that is what we actually do with real problems), the effectiveness of the algorithms will be approximately equal to the mean value on this task and not to the best value. Such a comparison on the tasks is shown in table 2, where two lines for each algorithm contain the values of the objective function, found in the best and average settings through all the settings and columns contain averaging over runs, and the standard deviation. Table 1 Adaptive algorithms comparison with other algorithms on the tasks Oliver30, Eil51 3-opt IWDs GA Adaptive GA ACO Adaptive ACO Oliver30 best 423,741 423,741 423,741 423,741 423,741 423,741 average 428,610 425,500 432,356 434,239 423,782 426,428 std deviation 7,4740 3,7022 13,2365 13,6875 0,1675 3,7653 Eil51 best 428,872 428,872 431,953 429,118 429,484 429,484 average 438,598 437,279 447,943 449,938 432,732 433,936 std deviation 5,0160 5,57137 7,5848 9,94143 3,11622 2,88955 137 Вестник СибГАУ. № 4(50). 2013 Examples of the ACO, IWDs, GA and adaptive GA behavior on task 0liver30 Table 2 Adaptive algorithms comparison with other algorithms with different parameters variants on the tasks Oliver30 and Eil51 3-opt IWD GA Adaptive GA ACO Adaptive ACO best average best average best average Oliver30 Best run 423.741 423,741 423,965 423,741 424,139 423,741 423,741 428,684 423,741 Average run 434.61 426,413 434,75 431,647 442,982 434,485 424,04 443,771 426,428 Standard deviation 11.931 3,19925 10,1741 11,3784 12,8611 12,7539 0,42986 11,7151 3,7653 Eil51 Best run 435,58 433,101 450,775 442,672 444,923 440,817 428,872 478,636 429,118 Average run 450,346 442,05 468,64 450,913 460,762 457,286 429,866 496,615 434,634 Standard deviation 8,59725 8,33447 11,6792 7,6221 10,7719 12,0611 0,74094 11,1774 3,38365 Both adaptive algorithms show good results on these problems as they outperform other algorithms with their settings giving average performance. Although on average adaptive GA cannot outperform other algorithms with their best settings and adaptive ACO cannot outperform conventional ACO with the best settings, one has to remark that we don't know beforehand which settings of the algorithm on the given task will be the best. There are 16 variants of ACO and 24 variants of GA and IWDs that means much extra efforts for the determination of these “best” algorithms before they could win adaptive GA. One can use the part of these efforts to improve results of the adaptive GA or adaptive ACO. Besides, if we deal with real-world problems, possible situation is the best settings absence. It means that there are different best settings in the different steps of problem solving. In such cases self-configuring algorithms bring much more advantages. Conclusion. In this paper we compared performance of some heuristic algorithms of combinatorial optimiza tion, such as 3-opt algorithm, intelligent water drops algorithm, conventional genetic algorithm and conventional ant colony algorithm with self-configuring (adaptive) genetic and ant colony algorithms. Our investigations demonstrate that adaptive algorithms are the effective methods of optimization with the remarkable property, which consists in the fact that the user does not have to adjust parameters but can have competitive results in solution quality. As a future work plans, it can be comparison with other algorithms, development of adaptive versions of other algorithms and using suggested approach for solving real world problems.

Об авторах

Ольга Евгеньевна Семенкина

Сибирский государственный аэрокосмический университет имени академика М. Ф. Решетнева

Email: oleese@mail.ru
бакалавр прикладной математики и информатики, магистрант кафедры системного анализа и исследования операций Российская Федерация, 660014, Красноярск, просп. им. газ. «Красноярский рабочий», 31

Евгений Александрович Попов

Сибирский государственный аэрокосмический университет имени академика М. Ф. Решетнева

Email: epopov@bmail.ru
доктор физико-математических наук, профессор кафедры системного анализа и исследования операций Российская Федерация, 660014, Красноярск, просп. им. газ. «Красноярский рабочий», 31

Ольга Эрнестовна Семенкина

Сибирский государственный аэрокосмический университет имени академика М. Ф. Решетнева

доктор технических наук, профессор кафедры системного анализа и исследования операций Российская Федерация, 660014, Красноярск, просп. им. газ. «Красноярский рабочий», 31

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