Stochastic Search Algorithms for Optimal ... - Semantic Scholar

Stochastic Search Algorithms for Optimal Monitoring Network Designs Ramiro Ruiz C. a, Marco A. R. Ferreira b,∗ and Alexandra M. Schmidt a a Instituto

de Matem´ atica,Universidade Federal do Rio de Janeiro, Brazil

b Department

of Statistics, University of Missouri - Columbia, USA

Abstract The design of efficient monitoring networks is very important to obtain a better understanding of environmental, ecological and epidemiological processes. In this paper we develop for the optimal design of monitoring networks a new hybrid genetic algorithm (HGA) which combines the standard genetic algorithm (GA) with a local search operator. We compare the performance of our HGA with two other stochastic search algorithms, a simulated annealing algorithm (SA) and a standard genetic algorithm (GA). Specifically, we consider the reduction of pre-existing largescale monitoring networks, when the optimality criterion is the maximization of the entropy of the included stations. The algorithms were tested on a set of simulated problems of different sizes, as well as on a real application involving the downsize of a large-scale environmental monitoring network. Even though the HGA spent more CPU time, it outperformed the other two algorithms in terms of solution quality for all the cases considered in the simulations as well as in the real application. Key words: genetic algorithms, simulated annealing, spatial sampling, ozone network design.

1

Introduction

The design of efficient monitoring networks is very important in order to obtain a better understanding of environmental, ecological and epidemiological ∗ Corresponding author. Email addresses: [email protected] (Ramiro Ruiz C.), [email protected] (Marco A. R. Ferreira), [email protected] (Alexandra M. Schmidt).

Preprint submitted to

2 August 2006

processes. The re-design of these networks, specially their downsize due to eventual future changes in priorities or objectives of the network as well as due to possible funding constraints, is another challenge that must be carefully addressed in order to minimize the loss of information and maintain the capability for inference. In this paper we develop a new hybrid genetic algorithm (HGA) for the optimal design of monitoring networks which combines the standard genetic algorithm (GA) with a local search operator. We compare the performance of our HGA with two other stochastic search algorithms: a simulated annealing algorithm (SA) and a standard genetic algorithm (GA). The focus is on the re-design of environmental large-scale monitoring networks. In general the design problem consists in determining a subset of sampling locations satisfying some optimality criterion. Many applications in the environmental monitoring network design literature consider the minimization of some cost function such as the Kriging variance (see for example Boer et al., 2002; Diggle and Lophaven, 2006; Pardo-Ig´ uzquiza, 1998; Trujillo-Ventura and Ellis, 1991, and references therein). Another approach based on information theory as proposed in Caselton and Hussain (1980) and Caselton and Zidek (1984), seeks to find a network design which maximizes the entropy of the random variables at monitored (gauged ) sites. The re-design of monitoring networks based on this criterion has been considered in Caselton et al. (1992); Guttorp et al. (1993) and Wu and Zidek (1992) for the univariate case, with extensions to a multivariate setting addressed in Le and Zidek (1994); Le et al. (2003) and Zidek et al. (2000). Independently of the design criterion used, computation is still a major practical consideration in spatial sampling design, as pointed out by Guttorp et al. (1993) and Le et al. (2003). When the space of possible solutions is finite and the criterion for evaluating the designs is well defined, the design problem may be viewed as a combinatorial optimization problem. The methods routinely used in the literature for finding the best solution to this problem consist of complete enumeration of all possible solutions (Le et al., 2003; Wu and Zidek, 1992); or simple heuristic rules, based mostly in exchanges, where at each step is added (or subtracted when reduction of the network size is being considered) the station that best improves the objective function to be optimized (Boer et al., 2002; Guttorp et al., 1993; Royle, 2002; Zidek et al., 2000); or exact algorithms which have been successfully developed for the Gaussian case (Bueso et al., 1998; Ko et al., 1995; Lee, 1998). Even though these methods work well for the design of small networks, they are not suitable for large networks. In fact, Ko et al. (1995) showed that this problem is NP-hard, implying that it is unlikely that an efficient algorithm can be found to solve in an optimal way all the instances of the problem. For this reason, monitoring network designs based on computational intensive stochastic search algorithms, such as the simulated annealing algorithm 2

(Banjevic and Switzer, 2001; Ferreyra et al., 2002; Ferri and Piccioni, 1992; Fuentes et al., 2005; Nunes et al., 2004; Pardo-Ig´ uzquiza, 1998) and the genetic algorithm (Reed et al., 2000; Wu et al., 2005), have recently been proposed in the literature. These algorithms have supplied good near optimal solutions to complex optimization problems in many areas of application. Along these lines we propose a new hybrid GA which improves the performance of the standard GA and SA algorithms. The performance of the HGA is compared with that of the SA and the standard GA algorithms on a set of simulated problems of different sizes, as well as on a real application involving the downsize of a large-scale environmental monitoring network. This comparison highlights the strengths and weaknesses of each algorithm. Entropy was chosen as the design criterion given its generality and wide use in the environmental literature. To our knowledge, the comparison of these stochastic search algorithms on a large-scale monitoring network design context has not been considered before in the environmental or statistical literature. A previous comparison of eight algorithms made by Lee and Ellis (1996), including SA and GA, was restricted to simple problems (their most complex problem was to choose 10 monitoring stations out of 121 possible candidate sites). The paper is organized as follows. Section 2 presents a review of entropy and information-based spatial sampling designs. Section 3 describes in detail the SA, GA and HGA algorithms for dealing with the problem of network design. Section 4 presents the comparison of the performances of our HGA algorithm and the SA and GA algorithms using simulated data. Section 5 applies the three algorithms to the reduction of a ground level ozone monitoring network in the United States. Finally, a short discussion and concluding remarks are presented in Section 6.

2

Entropy and Information-based Spatial Sampling Designs

Consider a random field, {Z(s) : s ∈ D ⊂

1 is an increase factor. Step 10 : Do c = c + 1 and return to step 4. We consider here a second order neighborhood scheme and a k-nearest neighbor scheme as neighborhood schemes for regularly and irregularly spaced designs respectively.

3.2 The Genetic Algorithm

The genetic algorithm (Holland et al., 1975) seeks to improve a population of possible solutions through generations using principles of genetic evolution such as natural selection, crossover and mutation. The implementation of the GA considered here consists of the following general steps: Step 0 : Initialize a counter c = 0 and specify some stop criterion (e.g., total number of generations totger). Step 1 : Choose at random an initial population of size N , that is, a set of N possible solutions S1 , . . . , SN . Step 2 : Calculate the fitness, that is, the value of the objective function H(Si ), i = 1, . . . , N , for each of the solutions in the population. Step 3 : Crossover. Choose a proportion, pcross , of solutions from the population. These solutions are selected according to a fitness-dependent selection scheme. Among these selected solutions, pairs of solutions at formed at random. From each pair a new solution is generated: Sites gauged in both solutions will automatically be gauged in the new solution; sites ungauged in both solutions will automatically be ungauged in the new solution; the remaining sites will be randomly assigned as gauged or ungauged respecting the total of n gauged sites. Step 5 : Mutation. Choose a proportion, propm, of solutions from the population with equal probability. For each selected solution, each gauged site may be swapped, according to a mutation probability p mut , with a randomly chosen ungauged neighbor site. Step 6 : Selection. The solutions generated by the procedures of crossover 6

and mutation are included in the population, and their objective function values (fitness) are calculated. The population of the new generation will be selected from this augmented population. The P solutions with better fitness (P < N ), called elite, will enter directly in the new generation while the other N − P members of the new generation will be randomly chosen according to a fitness-dependent selection scheme. Step 7 : Stop the algorithm if the stop criterion is met. Otherwise, do c = c+1 and return to step 4. We initially considered three fitness-dependent selection schemes: (i) a fitnessproportionate selection, where the probability of selecting a solution Si is given P by: p fitness = exp{H(Si )}/ N j=1 exp{H(Sj )}; (ii) a tournament selection where in order to select each solution, t solutions are randomly preselected (independent of the fitness) and the best out of the t is chosen; and (iii) a N + O selection, where the best N solutions are chosen from a temporary augmented population containing the N parents and the O offspring. For details of these and other selection schemes see for example Goldberg (1989). The best results were obtained for the second selection scheme with t = 4 solutions. Therefore, henceforth all the results are for the tournament selection scheme.

3.3 The Hybrid Genetic Algorithm

Our hybrid genetic algorithm couples the genetic algorithm of Section 3.2 with a local search algorithm (LS) in order to refine the quality of the solutions obtained by the GA, and consequently, to improve its performance. The general structure of the HGA proposed in this paper is described as follows: Step 0 : Initialize a counter c = 0 and specify some stop criterion (e.g., total number of generations totger). Step 1 : Construct an initial population of N solutions. Step 2 : Apply the local search procedure described in Section 3.3.1 to replace each initial solution with a local optimum solution. Step 3 : Apply the crossover and mutation operators to generate new offsprings. Step 4 : Select the N solutions for the population of the next generation as described in the GA algorithm. Step 5 : Apply the local search procedure to the solutions of the new population coming from the offsprings set. Step 6 : Repeat steps 3 to 5 until the stop criterion is met.

7

3.3.1 The local search operator The local search operator moves iteratively through the gauged stations sik of each solution Si , i = 1, . . . , N , k = 1, . . . , n. The local search operates in the neighborhood ∂ik of each gauged station sik : an ungauged neighbor site is selected with probability p neigh , and a proposed solution is formed by swapping the current and selected sites. The proposed solution substitutes the current solution only if it has a better fitness. In order to avoid to do local search on regions of the search space in which the global optimum is probably not located, the local search operator is typically applied only after a certain number of generations (e.g., the first 20%) of the GA. Starting from this point, local search is applied to every new solution, every g generations of the GA. The cost of this local search step is measured by the number of evaluations of the objective function performed by the local search operator, and its value is measured by its effect on improving the solution quality of the standard GA. In the next two sections we implement and compare these three algorithms on simulated as well as on real data sets.

4

Comparison of Algorithms

In order to evaluate the performance of the above algorithms, a simulation study was carried out, considering regular lattices of five different sizes (5 × 5, 10×10, 20×20, 30×30 and 50×50), which represent discretized random fields with 25, 100, 400, 900 and 2500 potential sampling locations respectively. The number of points to be selected in each problem corresponded to 20% and 40% of the potential sampling locations for the 5 × 5 and 10 × 10 grids, 10% and 20% of the potential sampling locations for the 20 × 20 and 30 × 30 grids and 5% and 10% of the potential sampling locations for the 50 × 50 grid. With the exception of the first two cases (5 × 5 grids), complete enumeration is impractical; our third most simple problem (to choose 20 monitoring stations out of 100 possible candidate sites) involves a search space of over 520 possible solutions. The focus of this study was on the comparison of the efficiency of the algorithms in finding sampling designs close to optimal. Thus, it was assumed that the distribution that approximates the process Z as well as its parameters were known a priori. More precisely, it was assumed that Z | Σ ∼ N (0, Σ), with Σij = exp{−φ k si − sj k}, and φ = 0.01, where k . k denotes the Euclidean distance. The entropy criterion was considered as a design criterion. Therefore, the aim was to maximize H(Z2 ) = 21 log | ΣZ2 |, with Z1 and Z2 8

as described in Section 2. The total number of iterations in each run of the SA algorithm was set to 50000 for all the problems considered. The control parameter (temperature) was initialized at 200 and reduced at every 100 iterations using a discount factor of α = 0.95. The re-starting step was implemented at every 1000 iterations using an increase factor of β = 1.5. The GA was run for 1000 generations, with a population of 100 candidate solutions, a probability of crossover p cross = 0.8, a probability of mutation p mut = 0.05, and an elite of P = 5 solutions. These values were previously obtained from a small case study implemented on a 10 × 10 grid. The HGA was run for 300 generations. We used a probability p neigh = 0.6, constant through generations, which was defined before the application of the algorithm through some exploratory runs. The local search operator was not applied in the first 20% of the GA’s generations. Starting from this point, local search was applied to every new solution in each generation of the GA. A second order neighborhood scheme was adopted in all the three algorithms. In order to compare the performance of the algorithms in finding the best sampling design, 100 realizations of each algorithm were obtained, beginning with different initial randomly selected starting designs, except for the last three cases where, for computational reasons, only 10 repetitions of each algorithm were obtained. All three algorithms used the same set of initial seeds for the random number generator to maintain comparability among the three algorithms. The effectiveness of the algorithms was expressed in terms of the objective function value and the frequency of finding good solutions. This was done for each of the problems considered. The computational time needed to execute the algorithms was also considered. The Ox programming language version 3.3 (Doornik, 2002) running under Windows XP was used to implement all the algorithms. The analysis was carried out on an AMD Athlon XP 3000+ model PC, at 2.1GHz and 512MB RAM. A summary of the results is given in Table 1 and in Figures 1 and 2. In Table 1 we report the maximum value of the objective function obtained in all runs for each problem and for each of the three algorithms, as well as its average and standard deviation values, the average run time for one run and the number of times the best solution found by all the three algorithms was obtained in each algorithm. Several conclusions can be extracted from the examination of these results: For small problems (5 × 5 grids), all the three algorithms were able to find the optimal solution in 100% of the times (for these simple cases the true optimal was known). For all the other problems considered, the HGA performed the best in terms of the number of times that the best solution found by all the three algorithms was recorded, as well as in terms of the average values of the objective function. The performance of the HGA was followed by that of the SA algorithm, while the worst results were for the GA. The differences 9

in performance between the GA and the other two algorithms becomes more accentuated as the problem size increases. The SA spent much less time than the GA or the HGA algorithms. This computational efficiency is also more evident for bigger problems. The solutions obtained with the HGA algorithm also had a smaller variability as shown in the box plots of Figure 1 and in the standard deviations of Table 1, followed very close by the SA. The GA’s solutions had the greatest variability. This result suggests that the SA and the HGA algorithms are less sensitive to the choice of starting points. This is an important characteristic because in general starting points are arbitrarily chosen. On the other hand, the GA apparently makes a good exploration of the search space, but it has a poor performance hill-climbing to better solutions. This drawback of the GA is overcome in the HGA with the inclusion of the local search operator. The best solutions found for each of the considered problems are presented in Figure 2. Our results coincide with those of Lee and Ellis (1996) who reported TABU search and SA as being more successful than GA for a monitoring network design problem, based on their frequency of finding the best solutions. The poor performance of genetic algorithms when compared to simulated annealing has also been referenced by Hurley et al. (1996) for radio link frequency assignment problems. On the other hand, Drezner and Wesolowsky (2003) found that GA outperformed SA in a facility location problem. According to Adler (1993), a high dependence on the crossover operator and the use of very low mutation rates could explain the premature convergence of the GA, leading to a stagnation as the population becomes more homogeneous. However, the GA was run with the best rates of crossover and mutation found in the exploratory analysis.

5

Application: Reduction of an Ozone Monitoring Network

The performance of the algorithms was also tested in an application that considered the choice of an optimal design for reducing a large network of ground level ozone monitoring stations in the United States. The data set considered for this application had been collected by the US Environmental Protection Agency (EPA) and consisted of hourly measurements of ozone concentration in parts per billion measured during the period 1994-2003. Approximately half of the monitoring stations were out of operation during the winter period and many of them were in operation just for some of the ten years considered. For that reason, we only considered for the analysis stations that were in operation for all the ten years, and took into account in each year only the summer season (June 21 to September 22). In addition, only measurements taken at 3pm were considered. This hour of the day was set because it concentrates the 10

highest levels of ozone and the lowest levels of missing values. Therefore, the data set actually used in the analysis consisted of a total of 671 monitoring stations scattered irregularly across the country, and each station provided 94 × 10 = 940 observations through time. In order to normalize and variance stabilize the data, we used the square root transformation. We used the sample covariance matrix after removing the temporal effect from the original data. The algorithms were implemented using the same design criterion and parameter settings as those of the previous section. The neighborhood scheme used in this application consisted of the 10 nearest-neighbors of each station. Five different sizes of the reduced network were considered: 67,134,201,268 and 336. The best design from 20 optimizations of each algorithm was considered. The results shown in Table 2 confirm the best performance of the HGA algorithm. The SA algorithm was also better than the standard GA for the biggest networks (268 and 336 stations). However, for small and medium networks (67, 134 and 201 stations), the standard GA outperformed the SA algorithm in terms of average solution quality and variability. The maps with the best subsets of stations found in each case are presented in Figure 3.

6

Conclusions

In this paper we have developed a hybrid genetic algorithm (HGA) for solving the problem of re-designing a monitoring network, incorporating a local improvement procedure into the standard GA. This algorithm was able to perform search over the subspace of local optima more efficiently. We have compared our HGA with the SA and GA algorithms using artificial problems of different sizes as well as on a real data problem. Computational results have shown that the hybrid algorithm performs very well on all the problems tested, outperforming the other two algorithms in terms of solution quality. Although our main aim was to reduce the size of an existing network, the algorithms presented here could be adapted to consider the case of augmenting the network. Moreover, changes in the objective function such as the incorporation of costs as proposed in Zidek et al. (2000), or the combination of the entropy criterion with a utility function giving priority to some desirable characteristics of the network design problem at hand (e.g. the capability for identifying levels of pollutants above certain threshold) as proposed in Fuentes et al. (2005), are also problems that can be solved by our HGA. All the algorithms implemented here were run sequentially. However, the GA and HGA algorithms are inherently parallel, that is, the solutions of the population can be evaluated simultaneously. Therefore, their computational times can be significantly reduced in a parallel computational environment. 11

Acknowledgements

This work is part of the Ph.D. research of Ramiro Ruiz under the supervision of M.A.R. Ferreira and A.M. Schmidt, with the support of CAPES. M.A.R. Ferreira was partially supported by CNPq, grant 402010/2003 − 5. A.M. Schmidt was partially supported by CNPq, grant 305579/2004 − 5. The authors are grateful to Jenise Swall for providing the ozone data.

References Adler, D., 1993. Genetic Algorithms and Simulated Annealing: A Marriage Proposal. In: IEEE International Conference on Neural Networks, Vol. 2, San Francisco, 1104–1109. Banjevic M. and Switzer, P., 2001. Optimal Network Designs in Spatial Statistics. In: Royal Statistical Society Conference on Spatial Modelling, Glasgow, 1–15. Bernardo, J.M., 1979. Expected information as expected utility. The Annals of Statistics, 7, 686–690. Boer, E.P.J., Dekkers, A.L.M. and Stein A., 2002. Optimization of a Monitoring Network for Sulfur Dioxide. Journal of Environmental Quality, 31, 121–128. Bueso, M.C., Angulo, J.M. and Alonso, F.J., 1998. A state-space-model approach to optimal spatial sampling design based on entropy. Environmental and Ecological Statistics, 5, 29–44. Caselton, W. F. and Hussain, T., 1980. Hydrologic networks: Information transmition. Journal of the Water Resources Planning and Management Division, A.S.C.E., 106 (WR2), 503–520. Caselton, W. F. and Zidek, J. V., 1984. Optimal monitoring network designs. Statistics & Probability Letters, 2, 223–227. Caselton, W.F., Kan, L. and Zidek, J.V., 1992. Quality Data Networks that Minimizes Entropy. In: P. Guttorp and A. Walden (Eds.), Statistics in the Environmental and Earth Sciences, Halsted Press, New York, pp 10–38. Diggle, P. and Lophaven, S., 2006. Bayesian Geostatistical Design. Scandinavian Journal of Statistics, 33, 53–64. Doornik, J.A., 2002. Object-Oriented Matrix Programming Using Ox, 3th edition, Timberlake Consultants, London. and www.nuff.ox.ac.uk/Users/Doornik.Ox programming. Drezner, Z. and Wesolowsky, G.O., 2003. Network design: selection and design of links and facility location. Transportation Research Part A, 37, 241–256. Ferreyra, R.A., Apeztegu´ıa, H.P., Sereno, R. and Jones, J.W., 2002. Reduction of soil water spatial sampling density using scaled semivariograms and simulated annealing. Geoderma, 110, 265–289. 12

Ferri, M. and Piccioni, M., 1992. Optimal selection of statistical units An approach via simulated annealing. Computational Statistics & Data Analysis, 13, 47–61. Fuentes, M., Chaudhuri, A. and Holland D.M., Bayesian entropy for spatial sampling design of environmental data. Mimeo Series N◦ 2571, (Institute of Statistics, North Carolina State University, Raleigh, NC, 2005). Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, Reading. Guttorp, P., Le, N.D., Sampson, P.D. and Zidek, J.V., 1993. Using entropy in the redesign of an environmental monitoring network. In: G.P. Patil and C.R. Rao (Eds.), Multivariate Environmental Statistics, North-Holland Elsevier Science Publishers, New York, pp 175–202. Holland, J.H., 1975. Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor. Hurley, S., Thiel, S.U. and Smith, D.H., 1996. A Comparison of Local Search Algorithms for Radio Link Frequency Assignment Problems. Proceedings of the 1996 ACM Symposium on Applied Computing, Philadelphia, 251–257. Kirkpatrick, S., Gelatt, C.D. and Vecchi, M.P., 1983. Optimization by Simulated Annealing. Science, 220, 671–680. Ko, C.W., Lee, J. and Queyranne, M., 1995. An exact algorithm for maximum entropy sampling. Operations Research, 43, 684–691. Le, N.D. and Zidek, J.V., 1994. Network designs for monitoring multivariate random spatial fields. In: J.P. Vilaplana and M.L. Puri (Eds.), Advances in Probability and Statistics, p. 191–206. Le, N.D., Sun, L. and Zidek, J.V., Designing Networks for Monitoring Multivariate Environmental Fields Using Data with Monotone Pattern. Technical Report # 2003-5, (Statistical and Applied Mathematical Sciences Institute, NC, 2003). Lee, Y.M. and Ellis, J. H., 1996. Comparison of algorithms for nonlinear integer optimization. Application to monitoring network design. Journal of Environmental Engineering, 122, 524–531. Lee, J., 1998. Constrained Maximum-Entropy Sampling. Operations Research, 46, 655–664. Nunes, L. M., Cunha, M. C. C., and Ribeiro, L., 2004. Groundwater monitoring networks optimization with redundancy reduction. Journal of Water Resources Planning and Management. 130, 33–43. Pardo-Ig´ uzquiza, E., 1998. Optimal selection of number and location of rainfall gauges for areal rainfall estimation using geostatistics and simulated annealing. Journal of Hydrology, 210, 206–220. Reed, P.M., Minsker, B.S., Valocchi, A.J., 2000. Cost effective long-term groundwater monitoring design using a genetic algorithm and global mass interpolation. Water Resources Research, 36, 3731–3741. Royle, J.A., 2002. Exchange algorithms for constructing large spatial designs. Journal of Statistical Planning and Inference, 100, 121–134. Trujillo-Ventura, A. and Ellis, J.H., 1991. Multiobjective air pollution moni13

toring network design. Atmospheric Environment, 25A, 469–479. Van Laarhoven, P.J.M. and Aarts, E.H.L., 1987. Simulated Annealing: Theory and Applications. Kluwer, Amsterdam, 187p. Wu, J., Zheng, C. and Chien, C.C., 2005. Cost-effective sampling network design for contaminant plume monitoring under general hydrogeological conditions. Journal of Contaminant Hydrology, 77, 41–65. Wu, S. and Zidek, J.V., 1992. An entropy-based analysis of data from selected NADP/NTN network sites for 19831986. Atmospheric Environment, 26A(11), 2089–2103. Zidek, J.V., Sun, W. and Le, N.D., 2000. Designing and integrating composite networks for monitoring multivariate Gaussian pollution fields. Applied Statistics, 49, 63–79.

14

SA

algorithm

HGA

HGA

SA

algorithm

GA

SA

algorithm

algorithm

SA

HGA

HGA

(b) 10 × 10 - 40

GA

SA

GA

SA

algorithm

algorithm

SA

HGA

HGA

(c) 20 × 20 - 40

GA

HGA

(f) 30 × 30 - 180

algorithm

(h) 50 × 50 - 250

GA

(e) 30 × 30 - 90

HGA

objective function objective function

−31.44 −31.48 −44.1

−18.05 −18.15 −121.5 −122.5

GA

algorithm

SA

(a) 10 × 10 - 20

GA

(d) 20 × 20 - 80

GA

objective function

objective function objective function

−11.11

15

Fig. 1. Box plots for the objective functions obtained for the simulation study.

objective function

−44.3 −44.5

(g) 50 × 50 - 125

−130 −132 −134 −136

−44.5 −45.5 −46.5

objective function objective function

−11.13 −11.15 −51.45 −51.60 −51.75

200

100

Y

100

60 20

50

40

Y

150

80

100 80 60

Y

40 20 20

40

60

80

100

20

40

X

60

80

100

50

X

150

X

(b) 10 × 10 - 40

(c) 20 × 20 - 40

100

150

200

250 50

150

250

0

50

150

X

500

(f) 30 × 30 - 180

0

100

Y

300 100 0 0

100

300

500

0

X

100

300

500

X

(g) 50 × 50 - 125

(h) 50 × 50 - 250

Fig. 2. Best network designs obtained for the simulation study.

16

250

X

(e) 30 × 30 - 90

500

(d) 20 × 20 - 80

Y

150

Y 0

X

300

50

0

50

Y

150 0

50

50

Y

100

150

250

200

(a) 10 × 10 - 20

100

200

(a) 67 stations

(b) 134 stations

(c) 201 stations

(d) 336 stations Fig. 3. Best network designs obtained for the reduction of the ozone monitoring network.

17

Table 1 Results obtained from the Simulated Annealing algorithm (SA), the Genetic Algorithm (GA), and the Hybrid Genetic Algorithm (HGA) for the simulation study.

grid

SA

n maximum

mean

GA sd

CPU

Nopt

maximum

mean

HGA sd

CPU

Nopt

maximum

mean

sd

CPU

Nopt

5×5

5

−1.8056

−1.8056

0

1.0

100

−1.8056

−1.8056

0

8.2

100

−1.8056

−1.8056

0

3.3

100

5×5

10

−6.3796

−6.3796

0

1.3

100

−6.3796

−6.3796

0

9.9

100

−6.3796

−6.3796

0

8.1

100

10 × 10

20

−11.1081

−11.1124

0.0052

2.7

56

−11.1081

−11.1210

0.0086

19.3

16

−11.1081

−11.1081

0

59.4

100

10 × 10

40

−31.411

−31.4178

0.0040

11.3

0

−31.4086

−31.422

0.0140

38.9

1

−31.4086

−31.4092

0.0006

363.2

53

20 × 20

40

−18.034

−18.0518

0.0082

14.4

0

−18.045

−18.0952

0.0327

69.1

0

−18.0315

−18.0352

0.0038

822.1

32

20 × 20

80

−51.471

−51.5214

0.0241

50.7

0

−51.532

−51.6271

0.0429

171.2

0

−51.4428

−51.4686

0.0111

5874.8

1

30 × 30

90

−44.001

−44.0549

0.0278

98.8

0

−44.154

−44.3191

0.0725

360.3

0

−43.9355

−43.9714

0.0165

13824.6

1

18

30 × 30

180

−120.87

−121.024

0.0711

394.2

0

−122.31

−122.527

0.1858

1285.3

0

−120.7810

−120.8411

0.0378

131315.1

1

50 × 50

125

−44.395

−44.4936

0.0481

229.8

0

−45.161

−45.5004

0.2706

1028

0

−44.3324

−44.3718

0.0301

49062.9

1

50 × 50

250

−129.93

−130.106

0.1066

1049.2

0

−135.33

−135.802

0.3101

3523

0

−129.7609

−129.8404

0.0504

472140.6

1

Nopt: number of times that the best solution found by the three algorithms was recorded; CPU: average computational time (in seconds) for one run; n: number of stations to be included; sd: standard deviation.

Table 2 Results obtained from the Simulated Annealing algorithm (SA), the Genetic Algorithm (GA), and the Hybrid Genetic Algorithm (HGA) for the real application. SA

n

GA

HGA

maximum

mean

sd

CPU

Nopt

maximum

mean

sd

CPU

Nopt

maximum

mean

sd

CPU

Nopt

67

43.357

42.503

0.5928

39.5

0

44.033

43.792

0.1510

167.0

0

44.276

44.176

0.0647

722.7

1

134

70.370

69.312

0.6808

190.8

0

71.275

70.819

0.2202

620.3

0

71.815

71.783

0.0336

8730.1

1

201

87.284

86.227

0.5221

493.7

0

86.963

86.402

0.3211

1526.0

0

88.902

88.845

0.0381

32324.2

1

268

95.703

94.826

0.4164

1028.7

0

94.579

93.691

0.5305

3045.7

0

97.094

97.089

0.0521

72109.5

1

336

96.809

95.711

0.5356

1881.3

0

94.983

93.760

0.7160

5444.2

0

98.935

98.875

0.0855

149160.2

1

19