In this paper, a mathematical programming and simulation method is used to estimate the number of weevils (Neochetina bruchi Hustache, and N. eichhorniae Warner) necessary to initialize the INSECT model which simulates the biological control of waterhyacinth by the weevils. The objective is to estimate the initial input values for the adult population so that the sum of the absolute differences between the observed and the simulated numbers of weevils is minimized. In general, the simulated values using the initial values obtained from the mathematical programming problem were within the 95% confidence intervals of the actual field observations. Also, in many cases, the simulation results indicated trends similar to those indicated by the field data in both timing and the numbers of weevils

Akbay, Kunter S.

English

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Marquette Universitye-Publications@MarquetteMechanical Engineering Faculty Research andPublications Mechanical Engineering, Department of1-1-1991Using Optimization and Simulation Techniques toEstimate Initial Weevil PopulationsKunter S. AkbayMarquette UniversityPublished version. Journal of Aquatic Plant Management, Vol. 29 ( January 1991): 21-24. Permalink.© 1991 Aquatic Plant Management Society. Used with permission.J. Aquat. Plant Manage. 29: 21-24 Using Optimization and Simulation Techniques to Estimate Initial Weevil Populations KUNTER s. AKBA¥ ' ABSTRACT In this paper, a mathematical programming and simu-lation method is used to estimate the number of weevils (NMc/utina bruchi Hustache, and N. eichhQrnUu Warner) necessary to initilialize the INSECT model which simulates the biological contro l of waterhyacinth by the weevils. T he objective is to estimate the initial input values for the adult population so that the sum of the absolute differences be-tween the observed and the simulated numbers of weevils is minimized. In general, the simulated values using the initial values obtained from the mathematical program-ming problem were within the 95% confidence intervals of the actual field observations. Also, in many cases, the simu-lation results indicated trends similar to those indicated by the field data in both timing and the numbers of weevils. Key words: Waterhyacinth, Neochetina, computer simula-tion models, mathematical programming. INTRODUCTION INSECT is a computer model developed for personal computers (Akbay et al. 1988) that simulates biological con-trol of waterhyacinth (Eichhornia cT(mipes (Solms.) Mart.) by the weevils. The long-term objective for development of the INSECT model is to predict Neochetillil. impacts on waterhyacinth. The INSECT model requires information about the initial weevil population to be input at the start of the simu-lation . However, the actual field data to specify the initial weevil levels are rarely obtainable. Howell et al. (1988) es-timated starting numbers for weevils by using the first three sampling periods from a site specific data set. and back-calculating to determine the number of individuals based on the cumulative day-degrees. Often multiple runs were made to arrive at reasonable starting numbers. Herein, as an alternative to the back-calculation method, the INSECT model will be incorporated in a mathematical programming approach to estimate the number of adult weevils which enter a field based on ob-served population values. METHODS Goal programming is a type of linear programming where the objective is to minimize the deviations from cer-tain goals (Charnes and Cooper 1961 ). For this , a separate 'Department of Mechanical and Industrial Engineering, Marquetle University, Milwaukee, Wisconsin 53233. R«eived for publication June 19, 1989 and in revised form June 28. 199(). J. Aquat. Plant Manage. 29: 1991. goal constraint must be formulated for each goal. A goal can be defined as a desired level of performance that one wants to achieve. The objective function will be minimized when all the goals are met. In the method described in this paper, the goals are the actual field data observations. The objective is to determine the initial weevil population so that the sum of the absolute difference between the observed counrs of weevils and the simulated values is minimized (McClendon et al. 1984). Mathematical Programming Formulation. In general, this mathematical programming problem can be defined as fol-lows: Let. J ~ Xj ~ aij ~ b, ~ d - ~ , d·· , the set of days for which comparisons will be made with the observed larvae/adults data (i is an element of this set), the set of permissible range of days which is defined as the time interval during which the adult weevils are expected to emerge or enter the field (j is an element of this set). number of adults emerging or entering the field on day j (unknown), total number of simulated larvae/adults on day i resulting from one adult entering the field on day j, number of total larvae/adults obsel>'ed (goal) on day i, amount that the total number of simulated larvae/adulrs on day i is below bi> and amount that the total number of simulated larvae/adults on day i is above b i . In order to minimize the sum of the absolute deviations between the observed values and the simulated values, the problem can be formulated as follows: Minimize: Z = d l - + d l + + d2- + d2' + ... + d m- + ( I ) d • m Subject to: allX l + a l2X2 + a2tXl + a22X 2 + Xl, X 2, . .. , Xn 2: 0 d l - , d2- , .•• , dm - 2: 0 dl ", d2 ~ ,···,dm ~2:0 + al!,XI1 + d l- - d l + = bl + a211XI1 + d2- ~ d2+ = ~ (2) 21 The solution to this problem would be the number of adult insects entering the field on permissible days in order to minimize the sum of the absolute differences between the observed weevil counts and the simulated values. If any of the deviational variables are positive in the final solution, that would indicate that the corresponding goals are not exactly met. cene'ral M ethodology. The initialization method using a mathematical programming approach has basically four phases: Phase /: In the first phase of the method, the INSECT model must be run several times to generate a large dalabase. In each run, the model is initialized with only olle adult on a permissible day. The range for permissible days must be input by the user. For example, if the user assumes that the adult weevils will emerge during the J u~ lian days 4 through 80 with increments of 4 days (i.e. on days 4, 8, 12, ... , 72, 76, and 80), then 20 runs will be made. The outpu t for weevil development from each run is stored in a large database. Phase 2: In the second phase, a smaller database is gener-ated to provide information for the goal constraint equa-tions (2) in the mathematical programming formulation. Basically, this information consists of the ali values (the total number of simulated larvae/adults on a held observa-tion day i resuhing from one adult entering the field on a permissible day j). Phase 3: In this phase, a linear programming algorithm is used to find the optimum solution to the problem. Phase 4: In the fina l phase, the INSECT model is initialized with the optimum solution and the results are plotted. At this point the user may decide to make additional runs with different permissible range and/or different field data . RESULTS AND DISCUSSION The data sets used to test the mathematical program-ming approach described in ule previous section are from Florida. T hese data sets were used during the initial field comparison studies for the INSECT model and the results based on the back~ca1culation method were summarized in Howell et al. (1988) . In this paper, for each field data set, the initial weevil population is estimated by using ule mathematical prog-ramming approach described above. Lake Alice 1976 Data. Initial conditions for the 1976 Lake Alice simulation runs were as follows: 22 initial plant biomass (kg/sq m): 0 .705, permissible range: January 4 through April 10 with increments of 4 days, adults (numberlsq ml based on mathematical programming results: 6.15 on January 4, 6.85 on January 8, 3.77 on January 20, 15.15 on March 21, and 0.17 on March 25. The goals in the mathematical programming formu lation represent the actual field observations for N. eichlwrnine adults. There are 52 field observations. The model results compared very well when the IN-SECT model was initialized with the values obta ined from the mathematical programming problem. In Figure 1, the INSECT model predictions for adults are p lotted against the mean and the 95% confidence intervals for the 1976 Lake Alice data. 9111y eight of 52 cases were not within the 95% confidence intervals. These occurred main ly during the month of March and December. Also, the sim ulation results generated by the INSECT model indicated trends similar to those indicated by the field data in both timing and numbers of adult weevils. During the second portion of the momh of December, the simulation results showed a smaller rate of decrease in the popu lation than suggested by the field data. North Florida 1986 Site "PP". Starting numbers for model simulations of water hyacinth and Neochetina spp. dynamics at site "PP" in North Florida were determined to be as follows: initial plant biomass (kg/sq m): 1.1 10, permissible range: January 4 through March 2 1 with increments of 4 days, adults (numberlsq m) based on mathematical programming results: 20.36 on March 5, and 14.90 on March 21. This site contained N. eichhorniae as well as N. bruchi, where N. eichlwrniae exceeded N. &ruchi by about IOta 1. The goals in the mathematical programming formulation rep-resent the actual field observations for N. eichhornim: ad ults. There are 10 field observations. Model predictions for adult N. eichhornitu and for N. bruchi were always within the 95% confidence intervals as ~, 'E I • '" ~JJ T I • ~ ,• 0 • ~ , ~ , .~ lTtTIP ~ 1 0 • ~ T " " t1 ~ ~rJITI : 11-, .J'"'-r-.LUf.2/" ! » 0 ~ m ~ ., , . .. = "' ,.. H' Juri"n D"y Figure I. Simulated adull N. nchJwm~ population plotted against the 95% confidence imervals for the 1976 Lake Alke data. Aquat. Plant Manage. 29: 199 1. .+------,,--,lr---,---l..----L.-' "-=..--..,~~~ o ~ ~ " ~ ~ ~ ~ ~ = ~ l~ ~ Julion If"" figure 2. Simulated adult N. nchJwrniae population plotted against the 95% confidence intervals ror the 1986 North norida site ~PP" data. seen in Figures 2 and 3, respectively. For adult N. eich/wr-nitu, population peaks between the field and simulation data sets were consistent both in terms of timing and the magnitude. During the end of September and the begin-ning of October, the simulated population for adult N. eichlwrnine was considerably lower than the actual field ob-servations during that period. However, the simulated val-ues were still within the 95% confidence interval. For adult N. brtuhi, the field data observations indicated a gradual increase in the population from April to the middle of June. However, the simulated adult N. bruchi population showed a constant pattern during the same period, and both the magnitude and the timing of the peak around Julian day 174 were missed. However, the simu-lated values were still within the 95% confidence interval. • • • • • lr --1I I"-- t-a M H ~ ~ M _ ~ _ = ~ ~ -Julion Day SQuth Florida /986 Site "CA". Starting numbers for model simulations of waterhyacinth and Neocheti1Ul spp. dynamics at the site were as follows: initial plant biomass (kg/sq m): 1.200, permissible range: January 4 through March 21 with increments of 4 days, adults (numberlsq m) based on mathematical programming results: 6.63 on January 4, 11.29 on February I, 7.52 on :\farch 5, and 38.21 on :\farch 9. Percent N. eichlwrnitu and N. bruchi were 62% and 38%, respectively. The goals in the mathematical programming formulation represent the actual field observations for the third instar Neocheti1Ul spp. larvae. There are 12 field ob-servations. As it can be seen in Figure 4, four of 12 cases were nOt within the 95% confidence intervals of the field data. However, the simulated values of third instar Neocheti1Ul spp. larvae for Julian days 55 and 328 missed the confidence intervals by a very small amount. T he simu-lated values for the first three field data observations were consistently lower. However, the population peak onJulian day 114 (April 24) as suggested by the field data was met by the model. The simulated values for the third instar Neochetina spp. larvae generally agreed with the actual field data observations during the months of May, June, July, August, September, October, and November. At the begin-ning of December, the model prediCled a peak in the pop-ulation which was not suggested by the field data. These resu lts are similar to those presented in Howell et al. (1988). Therefore, both methods can be used to in-itialize the weevil population for the INSECT model. Since both methods are still being developed, the purpose of this paper is not to make comparisons between them. Instead, '" •• ~ • 0 , -s " • • " rJ> j " I 1 ~ ,"~ ~ 0 ~ 1 ! 1/\ I.. • • M " H ~ ~ ". ,. •• = -'M -Julian Day Figure 4. Simulate:d third insla r NwcMtinQ. spp. larvae population plotu:d Figure ~. Simulate:d adult N. bruclli population plotted ag-.tinst th e: 95% against the 95% confidence intervals for the 1986 North Florida site "CA" confidence intervals for the 1986 Nonh florida ~ite "PP" data. data . J. Aquat. Plant Manage. 29: 1991. 23 the purpose is to describe the mathematical programming method and to summarize the results obtained from the initial comparison studies. It is also possible that both the method of back-calculating and the mathematical p rog-ramming approach can be used together to estimate the initial weevil popu lation. For example, the initial weevil population obtained from the mathematical programming approach can be modified by the method of back-calculat-ing in order to improve the match between the simulated and the actual field data observations. Finally, this approach illustrates how mathematical programming and simulation modeling can be used in a comple mentary man ner. The approach is general in na-ture and can be used for oilier insect models where the actual field data for initialization of the model is not avail-able. ACKNOWLEDGEMENTS The research entitled "Computer Simulation Modeling of Biological Control Agents of Nuisance Aquatic Plants" was originally cOntracted to the University of Southern 24 Mississippi (USM) under Contract No. DACW39-88-K-0052 with the U.S. Army Engineer Waterways Experiment Station (WES). Subsequently, a portion of the research was subcontracted to Marquette University. Technical monitor from WES of this study was ~h. R. Michael Stewart and from US M was Dr. Fred G. Howell . LITERATURE CITED Akbay, K. S., Wooten,j. W., and How~lI . F. G. 1988. ·Simulation Model of Wal~rhyacimh and 115 Biocontrol Ag~nu; R~porl I. Fim·Gc:n~n.· tion Mod~ l ,· T~chnical R~pon A·88·8, US Anny Engineer Wal~rways Experiment Station. Vicksburg, MiS5. Charn~s, A., and Cooper, W. W. 1961. Maoogtmml M ()(ku and IndwtritU A.pplications of LinMr Pragromming. Vol's. I and 2, Wil~y and Sons, N~w York, NY. How~IJ , F. G., Akbay, K. 5., and St~wart R. M. 1988. "Field Validation and Improvemenl$ to INSECT, a PerwnaJ Compul~r Simulation Mod~l," Proceedings of th~ 22nd Annual Meeting of Aquatic Plam Conlrol Research Program, Portland , OR. 160·168. McClendon, R. W., Brown, L. G., and Akbay, K. S. 1984. "A Goal Prog. ramming Approach for eSlima ting Inse(:t P~51 Migration and Initial Infl ux," Presemed al I h ~ I nstituL~ of Managcmem Sci~nc~s XXVI Meeting. Cop~nhag~n, Denmark. Aquat. Plant Manage. 29: 1991. 

Using Optimization and Simulation Techniques to Estimate Initial Weevil Populations

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Using Optimization and Simulation Techniques to Estimate Initial Weevil Populations

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