Matrix Based representation Genetic Algorithm for solving optical network design problem

Transcription

1 Matrix Based representation Genetic Algorithm for solving optical network design problem Mohammed S. A. Elsersy Mahmoud Marie Mohammed Zaki Abdelmagid Shehab Gamal El-Din Computers Engineering Dept. AlAzhar University,Egypt Abstract In this paper a Genetic algorithm is proposed as a basis for the required solution. A matrix is used as a representation which gives a better results than other previous methods. Accordingly, a model for the optical network design problem was created. More than one fitness function is used for measuring the fitness of the candidate solution. The genetic algorithm is applied as an optimization method for wide area optical network virtual topology design.. Matrix representation is applied for network design using Genetic Algorithms because matrix representation is more appropriate than string representation.the most straightforward topology representation is matrix representation. It is including capacity of the link so the matrix is asymmetric. No one can deny the familiarity of the matrix representation and the network in real life this could give you the essence of the connectivity and its source and destination from matrix shape. 1.Introduction GA is best suited for problems where good solutions have characteristics in common with the optimal solution. Therefore, it is essential to encode the problem with these features in mind. GA encodes a solution using a binary string where each region of the string defines certain traits of the solution. In general, the variables to be minimized are mapped to a binary string with continuous variables discrete to a given resolution. Problem Feasibility means it gives a good results in solving that problem. Tuning of the GA parameters to fit this representation includes these operators : a) Crossover operator (single-point crossover) b) Mutation operator (swap two elements in the matrix) c) Selection method are: Group Modified Roulette Wheel by ranges Tournament selection Random selection Roulette wheel Ranking selection d) Fitness function many options are suitable for fitness function such as : The delay of the traffic The cost of the link The utilization of the link 2.Problem description A model has been built to express the problem including the data known about it, constraints that controls it, variables needed and the objective functions[1].

2 3.The proposed Algorithm Parameters of GA There are two basic parameters of GA, crossover probability and mutation probability. Crossover probability determines how often will be crossover performed. If there is no crossover, offspring is exact copy of parents. If there is a crossover, offspring is made from parts of parents' chromosome. If crossover probability is 100%, then all offspring is made by crossover. If it is 0%, whole new generation is made from exact copies of chromosomes from old population (but this does not mean that the new generation is the same).crossover is made in hope that new chromosomes will have good parts of old chromosomes and maybe the new chromosomes will be better. However it is good to leave some part of population survive to next generation. Mutation probability determines how often will be parts of chromosome mutated. If there is no mutation, offspring is taken after crossover (or copy) without any change. If mutation is performed, part of chromosome is changed. If mutation probability is 100%, whole chromosome is changed, if it is 0%, nothing is changed. Mutation is made to prevent falling GA into local extreme, but it should not occur very often, because then GA will in fact change to random search. Population size says how many chromosomes are in population (in one generation). If there are too few chromosomes, GA has a few possibilities to perform crossover and only a small part of search space is explored. On the other hand, if there are too many chromosomes, GA slows down. Research shows that after some limit (which depends mainly on encoding and the problem) it is not useful to increase population size, because it does not make solving the problem faster. Another important parameter is selection.chromosomes are selected from the population to be parents to crossover. The problem is how to select these chromosomes. According to Darwin's evolution theory the best ones should survive and create new offspring. There are many methods how to select the best chromosomes, for example roulette wheel selection, Boltzman selection, tournament selection, rank selection, steady state selection and some others. In Roulette Wheel Selection parents are selected according to their fitness. The better the chromosomes are, the more chances to be selected they have. Imagine a roulette wheel where are placed all chromosomes in the population, everyone has its place big accordingly to its fitness function. In Rank Selection the previous selection will have problems when the fitness s differs

3 very much. For example, if the best chromosome fitness is 90% of the entire roulette wheel then the other chromosomes will have very few chances to be selected. Rank selection first ranks the population and then every chromosome receives fitness from this ranking. The worst will have fitness 1, second worst 2 etc. and the best will have fitness N (number of chromosomes in population). Situation before ranking (graph of fitness s),situation after ranking (graph of order numbers) After this all the chromosomes have a chance to be selected. But this method can lead to slower convergence, because the best chromosomes do not differ so much from other ones. Finally at Steady-State Selection is not particular method of selecting parents. Main idea of this selection is that big part of chromosomes should survive to next generation [2]. GA then works in a following way. In every generation a selected few (good - with high fitness) chromosomes is used for creating a new offspring. Then some(bad with low fitness)chromosomes are removed and the new offspring is placed in their place. The rest of population survives to new generation. Elitism is name of method, which first copies the best chromosome (or a few best chromosomes) to new population. The rest is done in classical way. Elitism can very rapidly increase performance of GA, because it prevents losing the best found solution. Recommendations are often results of some empiric studies of GA, which were often performed only on binary encoding. Some recommendation for using GA are: Crossover rate: Crossover rate generally should be high, about 80%-95%.(However some results show that for some problems crossover rate about 60% is the best). Mutation rate: On the other side, mutation rate should be very low. Best rates reported are about 0.5%-1%. Population size: It may be surprising, that very big population size usually does not improve performance of GA (in meaning of speed of finding solution). Good population size is about 20-30, however sometimes sizes are reported as best. Some research also shows, that best population size depends on encoding, on size of encoded string. It means, if you have chromosome with 32 bits, the population should be say 32, but surely two times more than the best population size for chromosome with 16 bits. Selection: Basic roulette wheel selection can be used, but sometimes rank selection can be better. for advantages and disadvantages. There are also some more sophisticated method, which changes parameters of selection during run of GA. Basically they behaves like simulated annealing. But surely elitism should be used (if you do not use other method for saving the best found solution). You can also try steady state selection. Encoding: Encoding depends on the problem and also on the size of instance of the problem. Crossover and mutation type depend on encoding and the problem. Applications of GA Genetic algorithms has been used for difficult problems (such as NP-hard problems), for machine learning and also for evolving simple programs. They have been also used for some art, for evolving pictures and music. Advantage of Gas is in their parallelism. GA is raveling in a search space with more individuals (and with genotype rather than phenotype) so they are less likely to get stuck in a local extreme like some other methods. They are also easy to implement. Once you have some GA, you just have to

4 write new chromosome (just one object) to solve another problem. With the same encoding you just change the fitness function and it is all. On the other hand, choosing encoding and fitness function can be difficult. Disadvantage of GA is in their computational time. They can be slower than some other methods. But with today s computers it is not so big problem. To get an idea about problems solved by GA, here is a short list of some applications: a) Nonlinear dynamical systems predicting, data analysis b) Designing neural networks, both architecture and weights c) Robot trajectory d) Evolving LISP programs (genetic programming) e) Strategy planning f) Finding shape of protein molecules g) TSP and sequence scheduling h) Functions for creating images GA Drawbacks There are some drawbacks for genetic algorithms a) Interdependencies of GA parameters b) global optimum & local optimum c) finely-tuned local search Problems to be considered are Single point crossover between string and matrix representation including its effect on speed of convergence If link matrix converted to be a single string (row by row) Then crossover operator will make results biased to the solution begins with the first row(s) If the link is unidirectional (not bidirectional) then link matrix is not symmetric.if the link form node A to node B is not the same capacity As the link from node B to node A then link matrix is not symmetric. If the reachability problem raised (a node is not accessible to any node in a certain individual) then use mend of the link matrix (each column must contain at least 1 in it) if there is a column(s) contains all zeros then add 1 in each column designs for mesh communication networks must meet conflicting, interdependent requirements. This sets the stage for a complex problem with a solution that targets optimal topological connections, routing, and link capacity assignments. These assignments must minimize cost while satisfying traffic requirements and keeping network delays within permissible values. Since such

5 a problem is NP-complete (one that has a solution in polynomial time but can only be solved by nondeterministic Algorithms), heuristic techniques must be used to handle the complexity and solve practical problems with a modest number of nodes. The heuristic methods used to design mesh networks include branch exchange, cut saturation, and Mentor algorithms. Another heuristic technique, genetic algorithms appears ideal to design mesh networks with the capability of handling discrete values, multiobjective functions, and multi constraint problems. Existing applications of genetic algorithms to this problem, however, have only optimized the network topology. They ignore the difficult sub problems of routing and capacity assignment, a crucial determiner of network quality and cost. Presenting a total solution to mesh network design using a genetic algorithm approach. Not only does our method optimize network topology, it also optimizes routing and capacity assignment. In the following design for a proposed communications network, genetic algorithms produced a solution that costs 9 percent less and has two-thirds the delay of a typical design method. Simulated Annealing versus Genetic Algorithm Simulated Annealing (SA) has been found to provide good solutions for complex optimization problems Both SA and GA can be used for solving NP-complete problems.sa has the difficulty of trapping in a local optima.but GA avoids this local optima by applying implicit parallel search technique. These differences that can be put in the following points: 1.Genetic Algorithm avoids the difficulty of trapping in a local optima which Simulated Annealing doesn t. While Genetic Algorithm deals with more than one solution at a time by implicit parallelism Simulated Annealing only deals with one candidate solution at a time. No information saved from previous moves to guide search but Genetic Algorithm saves best solution from generation to the next to by elitism. 2. When cost is used as a fitness function in determining the solution it has been found that cost is decreased by one third of its value. while the total capacity decreased only by 10% of its value also the delay decreased only by 10% of its value. 3. Applying different queueing models show that multiple processor with finite buffer decreased queueing delay While working with infinite buffer cut the delay by 7% of its value. 4. Genetic Algorithm operators were tuned for achieving the optimal solution,mutation probability with 0.05 gives better convergence. 5. Changing the demand of the network to its multiple raised the delay of the network linearly. Representation methods GA is best suited for problems where good solutions have characteristics in common with the optimal solution. Therefore, it is essential to encode the problem with these features in mind. GA encodes a solution using a binary string where each region of the string defines certain traits of the solution. In general, the variables to be minimized are mapped to a binary string with continuous variables discretized to a given resolution. Represents the solution in a matrix of N*N where N is the number of nodes in the problems. If there is a link between 2 nodes Lij = 1 Else Lij = 0 Number of 1 s is between N-1 and N(N-1)/2. The main diagonal of the matrix is zero because the traffic from node to itself is zero [2]. 4. Results The difficulty in developing a genetic algorithm to solve a particular optimization problem lies in the necessity of developing appropriate representation and encoding scheme for the solution space.the performance of a genetic algorithm heavily depends on solution representation,encoding scheme,and selection of genetic operators.

6 In this paper the matrix form is used for network representation because it is the most appropriate form in the network design problem. As in [3] the matrix is asymmetric because it is the more realistic case in which traffic is not the same in both directions [4]. Single point crossover operator is used to mix and recombine the solution to generate new off springs. Swap mutation is applied to improve performance by checking more solutions through randomization. Roulette wheel selection is applied for choosing the fittest solutions to be entered in the next generation.multiple fitness function are used as criteria in measuring the solution efficiency. Two case studies were tested the first was 10-nodes network in China as in, The second was the 14-nodes NSF network as in [1]. Effect of connectivity The various delay characteristic,overall delay, average propagation delay encountered by each packet, average queueing delay experienced by each packet,and the mean hop distance, as function milliseconds of the scale up ( throughput) shown in tables 3 to 6. the scale up provides an estimate of the throughput in the network. From these figures that the propagation delays is the dominant component of the packet delay. also,at light loads,the average propagation delay faced by packets in NSF net work is a little over 9 ms ( for the given traffic matrix), and this serves as a lower bound on the average packet delay. as a basis for comparison one-way propagation travels 40% of the coast-to-coast distance. Table 1 Maximum Capacity matrix for the case study

7 Fitness fn./ Parameter delay cost and only delay Cost(money unit) Delay(second) Capacity(Mbps) no of links Table 2 cost adding in fitness function results Table 3 Traffic matrix used for the case study A comparison held between the objective function considering delay only and.the objective function compromising cost and delay.results show that the total cost of the network is decreased sharply neither affecting the delay greatly nor the total capacity.only number of links reduced by four links. Also in virtual topologies all light paths were assumed to be routed over the shortest path on the physical topology, starting off with a random initial topology, a simple genetic algorithm was used to get the best virtual topology. The best virtual topology provided a maximum scale up of 126,clearly,the increased scale up demonstrates the benefits of the WDM-based virtual topology approach.now,the minimum loading was on link at 66% while all the other links were above 100% loading. The queueing delay is an important issue in the network design so a different queueing models is applied considering the physical topology and the virtual topology.first the simplest M\M\1 is applied to both cases.secondly more complicated model is used M\M\c with infinite buffer.this sounds unreal situation.the more practical model is M\M\c with finite size buffer.the queueing delay time is shown. To emphasize on the effect of genetic algorithm on the network design results a comparison with the branch exchange a classical method of design the results show that cost decreased but money save does not mean bad performance a lower delay is achieved with a higher capacity. To show the schema grow up with that objective function a genetic algorithm run and average cost decreased as generation number increased.

8 GA run average cost generation no. Figure 1 Average cost above for 100 generation With elitism Pm=0.1 From the figure 1 the fitness function grows as the generation goes up.figure 5 demonstrates different mutation rates affecting the fitness and the convergence of the problem as generations follows. Approach Genetic Algorithm Previous Genetic Algorithm * Branch Exchange * Total capacity (Mbps) Total cost (units) Delay (seconds) Table 4 Different queueing models results using GA. The algorithm has been applied to different network design cases applying a different fitness functions. the link costs in 10-nodes test case is obtained from [2], on the other hand in 14-node test case is obtained from real data taken from NSF network [1]. To test the performance of the algorithm under different settings, alter the value of crossover probability and the mutation probability,the costs of the best solutions in each iteration for different settings. Changing the mutation probability and its associated results is shown in fig. 5. Network design problem assume having a predicted traffic ( demand associated with each link) and cost of establishing that link in both directions. choosing single-point crossover & swap mutation to be applied on the network design problem. The effect of changing the mutation probability is shown in figure 2. Table 1 clarifies the impact of grouping the cost with the delay in the fitness function of the solution as a multi objective optimization problem. Meanwhile table 2 shows comparison of using WDM-in the network topology- between previous work [2] and our work, showing the Effect of variable demands versus delay is in figure 6. Table 6 shows the different queuing model used in the solving the problem with its result,only the M/M/1 is used in [1,2] the previous work. Table 7 summarizes the results for the previous genetic algorithm and branch exchange mentioned at [2] and our modified genetic algorithm. The results show the total capacity in Mbps,total cost by measured money units and finally delay in seconds.

9 Figure 2 Displaying the optimized network topology Test case for ten nodes optical network Sensitivity Analysis Effect of Nodal Degree Although no constraints on wavelengths per fiber imposed in this thesis, the wavelength requirements was examined to set up a virtual topology using shortest-path routing of light paths on the physical topology assuming no limit on the supply of wavelengths,but with the wavelength constraints.the maximum number of wavelengths required for embedding the best virtual topology ( which provided the maximum scale up) with nodal degree P= 4, 5 and 6 found to be 6, 8 and 8 wavelengths respectively. The corresponding distributions of the number of wavelengths used in each of the 21 fiber links of the NSFNET as in figure 6.10.With increasing nodal degree the number of the light paths to be supported increase and the average number of wavelengths a fiber needs to support increases. however,due to the combination of reasons such as desired virtual topology,shortest-path routing of light paths,and wavelength constraints,it may so happen that there is no link on the physical topology that employs all of the required wavelength. This happened when P=6 experiment, although eight wavelengths were required to embed the virtual topology,no physical link carried all eight wavelengths. Distribution of the number of wavelengths used in each of the 21 fiber links of the NSF network for the virtual topology approach with nodal degree P = 4,5 and 6. Effect of wavelength requirements The maximum scale up increased nearly proportionally with increasing nodal degree.actually,with the scale up of 126Mbps for P=4 as a baseline, proportional increase in scale up for P = 5 and 6 would yield 132.5Mbps and 159Mbps,respectively.However in these experiments the observed maximum scale ups P = 5 and 6 were higher 133Mbps and 163Mbps respectively.this is due to the fact that as the nodal degree is increased.the average hop distance of the virtual topology is reduced which provides the extra improvement in the scale up, minimizing hop distance can be an important optimization problem. Effect of network size Testing every possibility for an N city tour would be N! math addition,a 20 city tour would be 2.43*10^18 additions. assuming 2 billion additions per second.this would take over 22 years adding one more city would cause the numbers of additions to increase by a factor of 21. But

10 when GA is applied adding one more city increases the addition by population size multiplied by number of generations only which is reasonable by the same computing power. Effect of variable demand Variable demand means different demand. As the demand increases the delay of the network goes to the next level.after applying different traffic models it shown that delay increases fast with the increase in demand. The following traffic models are adopted.traffic model 1 : the number of light paths required between every pair of source-destination nodes is a randomly generated integer between 1and 5. Traffic model 2 : this traffic model is the same as traffic model 1. except that the demand matrix is symmetric.this model is used to study the influence of symmetric traffic. Traffic model 3 : the number of light paths required between every pair of nodes is a randomly generated integer between compared with the traffic model 1 and 2. this model specifies that the logical topology is sparsely connected but every logical connection between a pair of source-destination nodes has a larger mean number of light paths. the time complexities of the proposed algorithm and the other algorithms are very dependent on the cost matrix and the demand matrix. 5.Conclusion In this paper Genetic Algorithm is used to solve ten city network design problem, The total capacity has been increased by more than twenty Mbps, while the cost is decreased by 5%,and delay is reduced to half of its value. 1. When cost was added as a fitness function in determining the solution it has been found that cost is decreased by one third of its value. while the total capacity decreased by 10% of its value only. Almost the delay remains the same in both cases. 2. Applying different queueing models shows that multiple processor with finite buffer decreased queueing delay While working with infinite buffer cut the delay by 7% of its value. 3. Changing the demand of the network to its multiple raised the delay of the network linearly. 4. Genetic Algorithm operators were tuned for achieving the optimal solution,mutation probability with 0.05 gives better convergence. 5. The propagation delay is the dominant component of the packet delay. Also, at light loads, the average propagation delay faced by packets in NSF network is a little over 9 ms ( for the given traffic matrix ), and this serves as a lower bound on the average packet delay. The average queuing delay increase slightly with increasing traffic until the scale up nearly reaches its maximum value. The future work includes the cases that nodes locations is not given but the possible candidates for the cities (node locations) is given. So the algorithm must be changed to be efficient In solving that case. More General queueing models should be applied such as M/D/N/K and G/D/N/K to cover the general case in queueing models. The same representation could be applied on similar problems like process scheduling in multiprocessors environment.

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