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К оптимальному проектированию многослойных панелей On optimal design of multilayered panels

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phenomenon. Their study basically uses an important model of network influence largely due to DeGroot who studied consensus problems in groups of experts who originated in statistics [14]. In their paper, Golub et al. examined one aspect of this broad theme: for which social network structures will a society of agents who communicate and update naïvely come to aggregate decentralized information completely and correctly. They focus on situations where there is some true state of nature that agents are trying to learn and each agent’s initial belief is equal to the true state of nature plus some idiosyncratic zero-mean noise. The network structure of agents is described using a weighted network. Agents have beliefs about some common question of interest— for instance, the probability of some event. At each time step, agents communicate with their neighbors in the social network and update their beliefs. The updating process is simple. An agent’s new belief is the average of his or her neighbors’ beliefs from the previous period. An outside observer who could aggregate all of the decentralized initial beliefs could develop an estimate of the true state that would be arbitrarily accurate in a large enough society. Golub et al. studied learning in a setting where agents receive independent noisy signals about the true value of a variable and then communicate it in a network. The agents naïvely update beliefs by repeatedly taking weighted averages of neighbors’ opinions. They show that all opinions in a large society converge to the truth if and only if the influence of the most influential agent vanishes as the society grows. They also identify obstructions to this, including prominent groups, and provide structural conditions on the network ensuring efficient learning. Whether agents converge to the truth is unrelated to how quickly consensus is approached. The consensus problem is also related to synchronization. Synchronization is the most prominent example of coherent behavior, and is a key phenomenon in systems of coupled oscillators as those characterizing most biological networks or physiological functions. Synchronous behavior is also affected by the network structure. The continuous range of stability of a synchronized state is a measure of the system’s ability to yield a coherent response and to distribute information efficiently among its elements, while a loss of stability fosters pattern formation. In recent studies, the reason for the occurrence of synchronized networks became clear and the underlying network topology turned out to be important. However, synchronization often occurs unexpectedly and little is known what the best network topology is for synchronization. III. OPTIMAL NETWORK DESIGN FOR BETTER CONSENSUS The analysis of consensus problems relies heavily on matrix theory and spectral graph theory [15]. The interaction topology of a network of agents is represented using an undirected graph G with the set of nodes and edges. Neighbors of agent is denoted as . Consider a network of agents with the following dynamics: ̇  where ∑ ( ( ) ( ))   is the weight of agent on agent . Here, reaching a consensus means asymptotically converging to the same internal state by way of an agreement characterized by the following equation:    Assuming that the underlying graph G is undirected ( for all , j), the collective dynamics converge to the average of the initial states of all agents:  ∑  ( ) The dynamics of system in (1) can be expressed as  ̇ ( )  is the graph Laplacian matrix of the network G; the Laplacian matrix is defined as    where D = diag( ) is the diagonal matrix with ∑ elements and A is the binary adjacency matrix (n ×n matrix) with elements for all , where is 1 if agent and agent is connected or 0 if they are disconnected. 34 | P a g e www.ijacsa.thesai.org (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 5, No. 8, 2014 Notice that because the networks in this paper are undirected, L is a symmetric matrix with all real entries, and therefore a Hermitian matrix. In this research, L being a Hermitian matrix is always met with equality since the diagonal entry of each row in L is the degree of node , and each link connected to results in in the same row. So the sum of all off diagonals in a row is . Therefore L is a positive semi-definite matrix. Since L is semi-definite (and therefore also Hermitian), the following ordering convention for the eigenvalues will be adopted:    The interval in which the synchronized state is stable is larger for a smaller ratio of the two eigenvalues , therefore a network has a more robust synchronized state if the eigenvalue ratio is smaller. Focusing on the first part, the network optimization will be the guided evolution of networks subject to some constraints. Attempting to explain the formation of a small-scale size of networks, the constrained evolution of networks towards consensus or synchrony optimality will be investigated. Constraint optimizations are included via a fitness function that combines the desired goal such as synchronization properties of the networks optimized, or the propensity of a network to synchronize with an average degree requirement needed to connect the nodes. Ramanujan graph is known in the literatures as the best networks for fast consensus. Ramanujan graphs [16] are krandom regular networks with the second minimum eigenvalue satisfying:  ( ) √ ,  It is known that among the second minimum eigenvalue ( ) of various Laplacian matrices, Ramanujan graphs have the largest ( ) [17]. One class of Ramanujan graphs is a random regular network, which is easy to construct and used in many application for better consensus. optimization as a selection pressure to minimize the following fitness function is considered [18]:  ( ) ( )〈 〉 where is the average degree, and parameter controlling two objects. (  ) is a The eigenvalue ratio decreases as the average degree increases and the convergence speed becomes much faster. Therefore an interesting question is how to design a sparse network with a few degrees that guarantees a certain convergence performance. However this is a very difficult combinatorial problem. Therefore, an evolutionary design method is an effective way to design such a sparse optimal network. Initially 10 random networks with the Poisson degree distributions are generated and the genetic algorithm to obtain better networks in terms of improving the fitness function in (8) is used. The network is encoded as a binary adjacency matrix to perform the mutation and crossover. Next, the most suitable matrices among the parents and children matrices are chosen, and the others are eliminated. The multi-point crossover was used. After the crossover, each element in the matrix switches to a reverse state with a specific probability. In this paper, the network is an undirected graph, and so, if one element is reversed, the symmetry element is reversed at the same time. There is a possibility that an isolated network appears after crossover and mutation. In this paper, when an isolated node appears in a new network that the node has infinite distances to another node, the network is dumped. Therefore, non-isolated matrices can be used. After many generations have passed, an optimal network which minimizes the fitness function defined in (8) can be obtained. For dense networks with the average degree is larger than 4, the evolutionary optimized networks are Ramanujan graphs. However, for sparse networks with lower average degree, the condition in (7) does not carry any information especially for k=2. In this case a Ramanujan graph is a ring network with the degree of 2 as shown in Fig.1(a). However, the second minimum eigenvalue ( ) of the ring network is very small, and consensus is very slow on the Ramanujan graph with the degree k=2. However for a sparse network of the average degree =2, an evolutionary optimized network is a ring-trees type, where many modules networks with tree structures are combined by a ring network. Fig. 1 illustrates the difference of the network topology of an evolutionary optimized network (Fig. 1(b)) from a Ramanujan graph (ring network with k=2) (Fig. 1(a)). Genetic algorithms have been extensively used in single objective optimization for various communication network related optimization problems. Optimizing complex networks usually involve multiple objectives such as the network size as well as various network properties. In this paper, an evolutionary algorithm involving minimization of the eigenvalue ratio of the Laplacian matrix with the constraint of the average degree is used in order to design the optimal network. In this section, simulation results on fastest and slowest consensus formation on an evolutionary optimized network and a heuristically designed network are 189.5 190.5 190.8 188.8 191.4 189.2 194 187.5 187.8 187 186.8 186.4 186 184.8 186 182.8 181 178 180 180.2 180.5 183 183.8 180 189.5 195 (LPS) 0 1.27 0 0.607 2.469 3.517 2.063 1.031 0.55 1.27 0.989 2.61 2.259 1.34 1.059 0.211 0.382 1.186 2.116 1.27 0.564 4.515 8.903 10.055 1.91 2.364 2.939 10.309 11.145 3.901 0.91 1.673 1.27 - Base Demand (m3/h) 0 4.572 0 2.1852 8.8884 12.6612 7.4268 3.7116 1.98 4.572 3.5604 9.396 8.1324 4.824 3.8124 0.7596 1.3752 4.2696 7.6176 4.572 2.0304 16.254 32.0508 36.198 6.876 8.5104 10.5804 37.1124 40.122 14.0436 3.276 6.0228 4.572 - Shibu and Reddy 315 Table 13. Commercially available pipe diameters and unit cost of pipe for Bengali camp zone WDN inch mm 1 4 100 Unit Cost of Pipe (`/m length) 860 2 3 4 5 6 7 8 9 10 11 12 6 8 10 12 14 16 18 20 24 28 32 150 200 250 300 350 400 450 500 600 700 800 1077 1374 1840 2333 2885 3442 4142 4826 6375 8141 10161 Available Pipe Diameter Sl. No. 1) Model Run and Output for Case Study III At the start of the run, it is assumed that all the candidate diameters have equal probability of selection (i.e., P0,r=1/12). The performance function used for solving the model is given by (13). As the iteration progresses, some of the candidate diameters become superior to the others based on the performance values and their probability increases, while for others the probability gets reduced. This step-by-step iterative procedure for updating the probability of selecting a candidate diameter for each pipe will continue until they satisfy the stopping criteria. At the end, the probability of selecting a option for a pipe will be approximately equal to ones and zeros. This means that only 38 decisions (i.e., total number of pipes) will be having probability equal to one which forms the optimal solution set, and the remaining will be having a probability equal to zero. The stopping criteria is arrived in 38,400 objective function evaluations with smoothing parameter α = 0.35 and PN =108. The output of the model run for Bengali Camp Zone WDN is given in Table 14. Table 14. Cross Entropy Model Output For Bengali Camp Zone WDN Pipe No. Optimum Pipe Diameter (mm) Pipe No. 1 600 20 2 600 21 3 600 22 4 300 23 5 100 24 6 150 25 7 100 26 8 100 27 9 500 28 10 500 29 11 500 30 12 450 31 13 450 32 14 400 33 15 400 34 16 350 35 17 400 36 18 350 37 19 300 38 Optimum Cost (`) Optimum Pipe Diameter (mm) 200 100 100 150 200 100 100 100 150 150 100 100 150 150 100 100 100 300 150 25235630 Node No. Available Nodal pressure (m) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 10.99 11.8 17.43 18.19 18.38 14.74 16.52 20.12 16.21 15.11 14.82 16.65 14.19 16.27 11.58 18.38 18.10 Node No. Availabl e Nodal pressure (m) 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 107 18.89 19.05 19.43 19.78 20.86 19.57 22.55 24.29 27.20 25.13 24.90 24.48 21.94 21.49 25.00 15.97 18.89 - On comparing the results of the CE method for Bengali Camp zone WDN with the existing design, it is noticed that the optimal solutions of CE is better than existing design, resulting in 1.94% lesser cost than the existing design. The solution is obtained in 38,400 function evaluations. Also the minimum nodal pressure requirements are well satisfied. The results of present study amply demonstrate that the CE method is an effective optimization method for WDN and has capability to handle larger number of discrete decision variables and various constraints. Thus, CE method is well suited for optimal design of larger water supply networks. V. Conclusions This study presented Cross Entropy (CE) method for solving water distribution network optimization problems. For hydraulic simulation of WDNs, EPANET tool kit is adopted and carried out simulation-optimization modeling for design of WDNs. Initially, the CE method is applied for two benchmark WDN design problems, namely Hanoi WDN and Newyork city tunnel WDN. To evaluate the performance of CE optimization method, the results are compared with the past studies and it is found that the CE method is giving good quality optimal solutions in a few number of objective function evaluations. The results also demonstrated that the CE method can be used effectively for optimal design of new WDN as well as for rehabilitation of existing WDN (i.e., for capacity expansion of WDNs, in terms of adding parallel pipes without disturbing the existing pipes). It is also found that the CE method is capable of handling larger number of discrete decision variables and different types of constraints. After successful validation to standard WDNs, the CE method is applied to a real WDN in India and the results are compared with the existing solutions. It is found that CE method is giving minimum cost solutions (i.e., good quality optimal solutions) in quicker time (i.e., rapid convergence to optimum). Thus, the study concludes that the cross entropy optimization method is an effective optimization method for solving WDN problems, and which can be applied for optimal design of any practical WDN problems. References Gupta, I., Gupta, A., and Khanna, P., 1999, “Genetic algorithm for optimization of water distribution systems,” Environmental Modelling & software, 24(4), pp.437-446. [2] Goldberg, D.E., and Kuo, C.H., 1987, “Genetic algorithms in pipeline optimization,” Journal of Computing in Civil Engineering, 1(2), pp. 129-141. [3] Simpson, A. R., Dandy, G. C., and Murphy, L. J.,1994, “Genetic algorithms compared to other techniques for pipe optimization,” J. Water Resour. Plang. and Mgmt., ASCE, 120(4), pp. 423-443. [4] Savic, D.A. and Walters, G.A., 1997, “Genetic algorithms for least-cost design of water distribution networks,” J. Water Resour. Plang and Mgmt., ASCE, 123(2), pp. 67-77. [1] Cross Entropy Optimization for Optimal Design of Water Distribution Networks [5] Dandy, G.C., Simpson A.R., and Murphy L.J., 1996, “An improved genetic algorithm for pipe network optimization,” Water Resour. Res., 32(2), pp. 449 - 458. [6] Wu, Z. Y., and Simpson, A. R., 2001, “Competent genetic-evolutionary optimization of water distribution systems,” Journal of Computing in Civil Engineering, 15(2), pp. 89-101. [7] Cunha, M.D.C., and Sousa, J., 1999, “Water distribution network design optimization: simulated annealing approach,” J. Water Resour. Plang. and Mgmt., 125(4), pp. 215-221. [8] Eusuff, M. M., and Lansey, K.E., 2003, “Optimization of water distribution network design using the shuffled frog leaping algorithm,” J. Water Resour. Plang. and Mgmt., ASCE, 129(3), pp. 210-225. [9] Maier, H. R., Simpson, A.R., Zecchin, A. C., Foong, W.K.,Phang, K.Y., Seah, H.Y., and Tan, C.L.,2003, “Ant Colony Optimization for Design of Water Distribution Systems,” J. Water Resour. Plang. and Mgmt., ASCE, 129(3), pp. 200-209. [10] Shibu, A. Reddy, M.J., 2011, “Least cost design of water distribution network by Cross entropy optimization,” World Congress on Information and Communication Technologies (WICT), vol.,no., pp.302-306,11-14Dec.2011 doi: 10.1109/WICT.2011.6141262 [11] Rubinstein, R.Y., 1997, “Optimization of computer simulation models with rare events,” European Journal of Operations Research, 99, pp. 89-112. [12] Shannon, C.E., 1948, “A Mathematical theory of communication,” Bell System Tech. Journal, 27, pp. 379 423. [13] Kullback, S, and Leibler, R.A., 1951, “On information and sufficiency,” Ann. Math. Statics, 22, pp. 79-86. [14] Fujiwara, O., and Khang, D.B., 1990, “A two-phase decomposition method for optimal design of looped water distribution networks,” Water Resour. Res., 26(4), pp. 539-549. [15] Dijk, M. V., Vuuren, S. V., and Van, Z., 2006, “Optimizing water distribution systems using a weighted penalty in a genetic algorithm,” ISSN, Water SA, 34(5), pp. 378 - 478. [16] Vairavamoorthy, K., and Ali, M., 2000, “Optimal design of water distribution systems using genetic algorithms,” Computer-Aided Civil and Infrastructure Engineering, Blackwell, 15(4), pp. 374–382. [17] Schaake, J. and Lai, D., 1969, “Linear programming and dynamic programming applications to water distribution network design,”, Rep. No. 116, Dept. of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Mass. Author Biographies 1 Shibu A. is Research Scholar in the Department of Civil Engineering, Indian Institute of Technology Bombay, India. He completed his M.Tech.(Hydraulics Engineering) in 2000 from University of Kerala, India. His field of research is water distribution system modeling under uncertainty by using evolutionary techniques. 2 M. Janga Reddy is Assistant Professor in the Department of Civil Engineering, Indian Institute of Technology, Bombay, India. He 316 completed his Ph.D. in 2006 from Indian Institute of Science, Bangalore, India. His research interests include water resource systems: development of simulation and optimization models; optimal operations of single and multi reservoir systems; irrigation; hydropower; flood control; water distribution systems; multi-criterion decision making.
К Оптимальному Проектированию Многослойных Панелей On Optimal Design Of Multilayered Panels
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