1-s2.0-S0308521X01000257-main

8/14/2019 1-s2.0-S0308521X01000257-main

1/15

Robust parameter settings of evolutionary

algorithms for the optimisation of agricultural

systems models

D.G. Mayer a,*, J.A. Belward b, K. Burrage b

aQueensland Beef Industry Institute, Department of Primary Industries, LMB 4, Moorooka

Queensland 4105, AustraliabCentre for Industrial and Applied Mathematics and Parallel Computing, University of Queensland,

Queensland 4072, Australia

Received 21 July 2000; received in revised form 7 December 2000; accepted 31 January 2001

Abstract

Numerical optimisation methods are being more commonly applied to agricultural systems

models, to identify the most profitable management strategies. The available optimisation

algorithms are reviewed and compared, with literature and our studies identifying evolu-

tionary algorithms (including genetic algorithms) as superior in this regard to simulated

annealing, tabu search, hill-climbing, and direct-search methods. Results of a complex beef

property optimisation, using a real-value genetic algorithm, are presented. The relative con-

tributions of the range of operational options and parameters of this method are discussed,

and general recommendations listed to assist practitioners applying evolutionary algorithms

to the solution of agricultural systems.# 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Optimisation; Model; Genetic algorithm; Evolutionary algorithm; Parameters

1. Introduction

Systems research is increasingly being used for the solution of large, complex

problems (Hammel, 1997), including agricultural systems. The steps of model defi-

nition, development, validation, and investigation have been well documented (Dent

and Blackie, 1979; Bratley et al., 1987). For models where complete factorial

0308-521X/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved.

P I I : S 0 3 0 8 - 5 2 1 X ( 0 1 ) 0 0 0 2 5 - 7

Agricultural Systems 69 (2001) 199213www.elsevier.com/locate/agsy

* Corrresponding author. Tel.: +61-7-3362-9574; fax: +61-7-3362-9429.

E-mail address:[email protected] (D.G. Mayer).

8/14/2019 1-s2.0-S0308521X01000257-main

2/15

enumeration of the search-space (i.e. testing every combination of possible man-

agement options) is infeasible, the next logical step in this process is often the iden-

tification of the optimal management strategy for the model. To achieve this, themodel needs to be integrated with an optimisation algorithm, so that the combina-

tion of management options which gives the best simulated performance (profit-

ability, gross margin, or some other economic or utility measure) can be identified.

It is then expected that this simulated management strategy will extend to the real-

world system being modelled, and also produce optimal profitability there.

The object of this paper is first to review and compare previous optimisation

studies on agricultural and other models. The operational parameters of the method

found to be most efficient (a real-value evolutionary algorithm) are then discussed.

Results are presented for the optimisation of a complex whole-property model,

using this method. General discussions and recommendations are then listed con-

cerning likely robust parameters of evolutionary algorithms, for the future efficient

optimisation of agricultural systems models.

2. Optimisation methods for agricultural systems models

The numerical optimisation of an agricultural systems model requires the model to

be integrated with one of the many available optimisation algorithms. Interestingly,

for a number of these methods it has been theoretically proven that they will converge

to the global optimum as the number of iterations approaches infinity. The simplex

algorithm has been proven convergent, on strictly convex functions in lower dimen-

sions (Lagarias et al., 1998). Using Markov chain theory on combinatorial optimisa-

tion problems, this property also holds for simulated annealing (van Laarhoven andAarts, 1987), genetic algorithms which incorporate elitism (Peck and Dhawan, 1995),

and evolution strategies (Ba ck and Schwefel, 1993). However, these results are prob-

ably mostly of academic interest, as they cover only a subset of potential optimisation

problems, and are unrealistic. If infinite iterations were truly possible, the optimum

would simply be found by complete enumeration. What is required from a practical

viewpoint is a method which is both robust (i.e. it will reliably converge to the global

optimum) and efficient (also achieving this within minimal computing time).

From practical studies, a number of possible optimisation methods have been

shown to be either inappropriate for the task of model optimisation, markedly

inefficient, or prone to converging to local optima (rather than the targeted global

optimum). These inappropriate methods include the hill-climbing (or gradient-type)methods (Fletcher, 1987), direct-search algorithms (Bunday, 1984), including the

Nelder-Mead simplex (Nelder and Mead, 1965), and the tabu search metastrategy

(Glover et al., 1993).

The remaining two general families of optimisation methods have proven valuable

in this field. Based on the annealing or cooling processes of metallurgy, simulated

annealing (Kirkpatrick et al., 1983; Corana et al., 1987), and its more efficient

extension of simulated quenching (Ingber, 1993), have been used to successfully

identify the economic optima of a range of agricultural systems. However, their

200 D.G. Mayer et al. / Agricultural Systems 69 (2001) 199213

8/14/2019 1-s2.0-S0308521X01000257-main

3/15

rates of convergence have been identified as a concern for some problems, which

then leads on to excessive run-times being required to achieve the optimum. The

second successful family of methods covers the evolutionary algorithms (includingevolution strategies, binary and real-value genetic algorithms, and their various

combinations). By modelling the processes of natural selection and evolution, these

methods have been shown to be robust and efficient across a range of problem types

(Ba ck and Schwefel, 1993; Michalewicz, 1996).

3. Comparisons of optimisation methods

Quite a number of comparisons between optimisation algorithms have been con-

ducted on a range of mathematical test functions, with varying results. In general,

the more modern optimisation methods (simulated annealing and evolutionary

algorithms) appear superior, however the other methods can and do perform well on

specific types of problems. These test function results, however, are only approxi-

mately applicable to the optimisation of management strategies in systems models.

Simulation models form one of the more difficult classes of optimisation problems

(Mayer et al., 1998a). They are typically of higher dimensionality, as each individual

management option to be optimised contributes an extra dimension to the search-

space. The response surface generated by models can be non-smooth, and include

discontinuities and cliffs. These occur when the agricultural system is pushed too

far (e.g. overstocking), so that it crashes (both biologically and thus economically).

The impact of any altered management strategy may take a number of years to settle

in, so the dynamic nature of these trends needs to be accounted for. Also, agri-

cultural models have been shown to contain multiple local (as opposed to global)optimal solutions (Mayer et al., 1996; Hammel, 1997).

Given the limitations of the hill-climbing methods when applied to these types of

systems, it is not surprising that relatively few applications have appeared in the lit-

erature. Roise (1990) used the conjugate directions method to optimise small to

moderately sized spatial models of forestry stands. Generally, though, gradient

methods have more been used as benchmarks against which alternate optimisa-

tion methods are shown to be superior, in the optimisation of model profitability

(Mayer et al., 1991, 1996; Hart et al., 1998) and in model calibration (Hendrickson et

al., 1988; Wang, 1991).

Similarly, direct-search algorithms have been used with mixed success in the opti-

misation of agricultural models. Mayer et al. (1991, 1996) showed the simplexmethod performed better than gradient methods on a dairy farm model, but found

both to be inferior to the more modern algorithms. Botes et al. (1996) found the

simplex to be more flexible and realistic than dynamic programming on a crop irri-

gation problem, and Parsons (1998) showed it to have approximately comparable

performance to a genetic algorithm in determining an optimal silage harvesting plan.

Whilst the tabu search strategy is efficient when dealing with spatial or temporal

allocation-type problems, it is ill-suited to the optimisation of systems models (Mayer

et al., 1998b). To our knowledge, it has yet to be applied successfully in this field.

D.G. Mayer et al. / Agricultural Systems 69 (2001) 199213 201

8/14/2019 1-s2.0-S0308521X01000257-main

4/15

Simulated annealing has been used successfully in a number of modelling studies.

Optimal groundwater remediation strategies were determined using this approach

(Kuo et al., 1992). This model had previously defeated hill-climbing methods, whichtended to repeatedly converge to sub-optimal solutions. Considering the harvesting

schedule for forestry blocks, Lockwood and Moore (1993) report simulated anneal-

ing applications covering moderate to large sized problems. Bos (1993) also used

simulated annealing to solve a forestry management problem, which had proved too

computationally intensive for the branch and bound method. On a temporal har-

vesting model of a prawn fishery, Watson and Sumner (1997) adopted simulated

annealing to avoid being trapped by the many local optima. The CERES-Maize

agricultural model was calibrated to USA data (Paz et al., 1999), also via this

method.

Of the family of evolutionary algorithms, binary genetic algorithms have seen most

use in the optimisation of agricultural systems. In horticultural planning, Annevelink

(1992) outlined a spatial/temporal allocation problem which was solved by a genetic

algorithm, after initial linear and dynamic programming approaches proved inade-

quate. Horton (1996) showed genetic algorithms efficiently solved a sheep genetics

model, identifying optimal solutions which hill-climbing methods repeatedly failed to

find. The annual grazing management plan, drafting weights, and fertilizer and sup-

plementation usage of a sheep farm were successfully optimised with a genetic algo-

rithm (Barioni et al., 1997). Parsons (1998) showed a genetic algorithm to be

marginally superior to the simplex method in optimising the silage harvesting plans of

a farm. Hart et al. (1998) found a hybrid approach, incorporating a genetic algorithm

with hill-climbing, to be the best for optimising a dairy farm model.

Direct comparisons of the optimisation methods on our series of studies are listed

in Table 1. This shows the relative success of each (measured as the percent of the

Table 1

Average identified optima (% of global optimum), and number of model runs (000, in brackets) to

achieve each of these, for investigated agricultural models and optimisation methods

Dairy farma,b Herd dynamicsc,d

Restricted Full

Dimensionality 16 20 40

Search-space 1023 1050 10100

Hill-climbing (quasi-Newton) 84.66 (0.6)

Direct-search (Simplex) 94.27 (0.4)Simulated annealing 100 (103) 99.12 (816) 98.64 (3443)

Simulated quenching 99.80 (3.7) 99.01 (165) 99.15 (1393)

Binary genetic algorithm 99.68 (10) 99.76 (143) 99.96 (2428)

Evolution strategy 99.74 (10 423)

Real-value genetic algorithm 99.95 (3456)

a Mayer et al. (1995).b Mayer et al. (1996).c Mayer et al. (1999a).d Mayer et al (1999b).


8/14/2019 1-s2.0-S0308521X01000257-main

5/15

global optimal profitability), as well as their efficiency (the number of individual

model runs to achieve this). The earlier investigations (Mayer et al., 1995, 1996)

confirm the hill-climbing and direct-search methods as practically unsuitable for thismodel. In the latter studies (Mayer et al., 1999a, b) all investigated methods per-

formed reasonably well. The obvious concern here (Table 1) is the notable de-

terioration of both the efficiency and reliability of simulated annealing and

quenching, as the problem size increases.

4. Beef property optimisation

Considering an average beef property in the northern speargrass region of

Queensland (ORourke et al., 1992), previous investigations of the herd dynamics

submodels (Mayer et al., 1999a, b) provided valuable insights into optimal trading

(buying and selling) decisions. Using an annual time step, the numbers of animals in

each sex and age cohort (Nsex,age) are modelled by:

Nsex;age Nsex;age-1 1Mortalitysex;age-1

Purchasessex;age Salessex;age;

with

Nsex;0 0:5X10

age2

Nfemale;age ProportionBredfemale;age PregnancyRatefemale;age

1CalfLossRatefemale;age Purchasessex;0 Salessex;0

However, the operation of a beef production enterprise encompasses a much wider

range of strategies than just trading decisions. To properly investigate the full range

of management options available to property operators, the model must be able to

realistically represent all the key processes and pathways of this system. No model

can totally simulate such an operation, but consultations with producers and

experienced personnel identified the key biological and managerial processes which

were likely to be of most significance. The herd dynamics submodel was enhanced to

allow a daily (rather than annual) time step, allowing within-year details such as

mating period, weaning dates, and buying and selling times. Also, the weights and

life-histories of individual animals were simulated (rather than just cohorts of

average animals), thus modelling the effects of delayed or out-of-season pregnan-cies on weight dynamics, calving, and future performance.

The key dynamic biological and managerial processes incorporated in this model

are as follows.

4.1. Reproduction rates

At each day within the defined mating period, each cow is given a stochastic

probability of conception, via a relationship based on breed, lactation status, age


8/14/2019 1-s2.0-S0308521X01000257-main

6/15

and condition (Mayer et al., 1999c). Following the (randomly determined) simulated

pregnancy of a cow, calving occurs deterministically 284 days later, with calves

having defined weights of 30 kg (Zebu crosses), 32 kg (Africanders), 34 kg (Britishbreeds) and 41 kg (European breeds). A stochastic calf mortality rate of 10% is

applied at birth.

4.2. Mortality rates

For males (bulls and steers), mortality rates in the target region are generally

assumed to be low, and of less importance than the higher mortalities of breeders.

Currently for male animals, stochastic mortality is applied at a rate of 2% per

annum, applied on a daily basis. Female animals (heifers and breeders) also experi-

ence stochastic mortality, dependent on age, condition, and rate of weight change

(Mayer et al., 1999c).

4.3. Weaning

Two weanings are allowed per year, at any nominated dates. The first weaning is

optional, and (if used) needs a nominated minimum weaning weight any calves

below this weight are not weaned at this time. The second weaning is mandatory,

and all calves are weaned at this date.

4.4. Herd and farm structure

A stable opening herd, where sufficient calves are retained each year to replace the

older animals dying or being sold off, was estimated via the annual modelBREEDCOW (Holmes, 1995). This herd was consistent with surveyed production

data (ORourke et al., 1992). A bull ratio of one bull per 30 breeders was maintained

by additional annual sales or purchases of bulls. Spaying of cows was not con-

sidered, and the breed used was nominally Zebu Fn

crosses (allowable alternate

breed types are Zebu F1 crosses, Africanders, British breeds, and Europeans).

Overall average current prices by animal type and age were estimated, from

National Livestock Reporting Service data. Variable per annum costs were $12 per

calf, $15 per bull and $6 for other animals, and fixed property costs were set at

$80 000 p.a.

Given the excessive run-times of this model on a personal computer, it was ported

across to SUN workstations running Unix, and recoded to become a subroutinecalled from within the evolutionary algorithm GENIAL (Widell, 1997). The objec-

tive function to be optimised was taken as the non-discounted operating profit over

10 years, plus the value of the closing herd, and less the value of the initial herd. Even

using the somewhat restricted and thus simplified version of this whole-property

model, there are 70 management options to be considered, as listed in Table 2. This

poses a very difficult problem to optimise.

The combination of problem size and lengthy computations remained a concern.

Depending on the workstation used, individual 10-year model runs took 3050 s


8/14/2019 1-s2.0-S0308521X01000257-main

7/15

each. Hence, even a one-month optimisation on the fastest machine would only

generate around 90 000 model runs, which is well short of the 29 million required

for convergence by the various optimisation methods in Table 1. Also, this current

model is far more complex than those previously tackled, with 70 dimensions and a

practical search-space of the order of 10120 possible combinations. So, only the most

efficient routines can be considered for this task, which rules out simulated annealing

and evolution strategies. The binary and real-value genetic algorithms performed

similarly on the largest problem tackled thus far (Table 1). For the optimisation ofthe whole-property model, the real-value genetic algorithm was selected. This was

mainly because it incorporates a wider range of operational parameters and options,

importantly including boundary mutation which is likely to enhance its perfor-

mance with this model (as many of the trading decisions, in particular, will be opti-

mal at their boundaries).

5. Operational parameters of evolutionary algorithms

Practitioners using evolutionary algorithms in the optimisation of management

strategies using agricultural systems models are faced with a wide range of opera-tional parameters. These control the balance between exploitation (using existing

genetic material in the population to best effect) and exploration (searching for bet-

ter genes). The operational parameters can be fine-tuned in practice, and have

varying degrees of effectiveness some operators may prove critical to efficiency,

with others less so. The key operators of recombination and mutation have a

synergistic effect (Fogel, 1995), so these should always be used in conjunction with

each other. In practice, evolutionary algorithms using only one of these have proved

inferior (Ba ck and Schwefel, 1993; Michalewicz, 1996). Overall performance of the

Table 2

Management options and bounds for the whole-property beef model

Number Definition Bounds

1 Starting day (of the year) for the mating period. 1364

2 Length of the mating period, in days. 1364

3 Is early weaning used? Yes or no

4 Day of the year for early weaning. 190

5 Minimum weight (kg) used in early weaning. 60120

6 Day of the year for late weaning. 91140

714 Number of steers (ages 18, respectively) bought in each year. 020

1522 Number of cows (ages 18, respectively) bought in each year. 020

2330 Day of the year for purchasing steers (ages 18). 120

3138 Day of the year for purchasing cows (ages 18). 120

3946 Proportion of steer age cohorts (ages 18) sold each year. 01

4754 Proportion of cow age cohorts (ages 18) sold each year. 01

5562 Day of the year for selling steers (ages 18). 141210

6370 Day of the year for selling cows (ages 18). 141210


8/14/2019 1-s2.0-S0308521X01000257-main

8/15

various operators remains likely to be problem-dependent (Goldberg, 1989; Fogel,

1995; Horton, 1996; Michalewicz, 1996). Practitioners must weigh up the relative

advantages and disadvantages of this range of operators, including the following.

5.1. Coding of modelled options to genetic representation

Within genetic algorithms, the two main options are real-value coding (where each

gene is a numerical representation of one of the management options being opti-

mised), and binary coding (where each option is mapped onto one or a series of

binary genes, dependent on the precision required for each). In practice, there

appears little difference in performance between these methods of coding (Mu hlen-

bein and Schlierkamp-Voosen, 1994), as was also evident in Table 1. Real-value

coding was selected for the optimisation of the whole beef property model, because

of the suggested superiority of a 1-to-1 option-to-gene coding (Michalewicz, 1996),

as well as the wider available range of potentially useful genetic operators for this

representation.

5.2. Population size

Each individual of the genetic algorithms population consists of a trial combina-

tion of management options. The population must contain a sufficient number of

individuals to maintain genetic diversity, but carrying too many proves inefficient.

Hart et al. (1998) found 10 to be insufficient, and a range of reported studies indicate

values of up to 200, with an overall average of about 50. However, the optimal value

for any given problem is obviously related to its size and dimensionality (Peck and

Dhawan, 1995). For the 16-dimensional dairy farm model (Mayer et al., 1995, 1996),a population size of 40 proved marginally better than 80. With the 20-dimensional

herd dynamics model (Mayer et al., 1999a), sizes of 30 and 50 performed approxi-

mately equally. For the 40-dimensional version (Mayer et al., 1999a, b), the binary

genetic algorithm showed populations of up to 100 to be slightly better than one

optimisation using 150. On this same model, the real-value genetic algorithm showed

a population of 100 converged more rapidly than 50 or 500, however, this comparison

was partially confounded with different mutation operators. Taken overall, it appears

that population sizes need to be (at least) greater than the dimensionality of the

studied problem. As there are understandable dangers of having too small a popula-

tion, and not much penalty for a few too many, a population size of twice the

dimensionality appears both quite efficient and intuitively safe.

5.3. Selection of parents

Given the range of possible selection schemes, it is fortunate that this operator

appears to be one of the less critical. The various types of selection schemes have been

shown to produce similar selection pressures across the population (De Jong and

Sarma, 1995; Blickle and Thiele, 1997). Roulette-wheel selection based on ranks or

scores (with a variety of possible scaling options) tends to be more computationally


8/14/2019 1-s2.0-S0308521X01000257-main

9/15

demanding, particularly when used with steady-state replacement. Conversely, tour-

nament selection can be readily and efficiently implemented with any of the other

operators. Overall results indicate that tournament selection, with a tournament sizeof two, performs well over a number of problem types. If more selection pressure is

desired, then truncation selection or a larger tournament size could be used.

5.4. Replacement strategy

Initial genetic algorithms tended to use a generation gap of one, whereby all par-

ents were replaced by the next generation of offspring. Under this scheme, elitism

(where, at least, the best parent is retained) was strongly recommended. More

recently, steady-state replacement has become more common, with only the poorest

individuals (or single individual) of the population being replaced at each step. This

ensures retention of the best parents, and also makes the new offspring immediately

available to the optimisation, which is advantageous as they should be amongst the

best individuals at any time. When adding the new individuals to the population,

the deterministic replacement of the worst parents seems logical, given that any sto-

chastic (random) replacement method can result in the loss of some of the elite parents.

5.5. Mutation

The binary genetic algorithms traditionally used mutation as a background

operator, having only low rates of the order of 0.001 to 0.01. Given that each mod-

elled real-world option here usually mapped to a string of binary genes, mutated

values of these options would occur at higher probabilities than these base rates, but

still only occasionally. This results in much longer escape time for populationswhich have converged (primarily via the process of recombination) onto local

optima. Genetic algorithms using low mutation rates can stay converged (within

available computational time) on these local optima, as mutation occurs too infre-

quently in practice to find the optimal region. More recently, and especially with

real-value codings, higher mutation rates (up to about 0.4) have been found bene-

ficial (Hinterding et al., 1995; Michalewicz, 1996; Ba ck, 1997). Mayer et al. (1999b)

list good results from quite a wide range of mutation probabilities, up to 0.6. In

addition, multiple mutation (incorporating at least two different types of mutation

operators, at varying probabilities) also appears more successful. The choice of just

which mutation operators to use is also important on real-value genes, uniform

or Gaussian creep (in the style of evolution strategies) appear more successful thantraditional uniform (random) mutation, and boundary mutation can be very

effective on problems where the global optimum occurs when a number of the input

options lie at their respective boundaries.

5.6. Recombination

Often considered to be the driving force of evolutionary algorithms, recombination

(or crossover) has a wide range of possible types, namely one-point, multiple-point,


8/14/2019 1-s2.0-S0308521X01000257-main

10/15

uniform, intermediate arithmetical, and extended arithmetical. These have been

applied across the whole range of possible probabilities (0.001 to 1.0), generally with

reasonable success. In our research, all types (except intermediate arithmeticalcrossover) have been used with good effect, and here there is no indication of the

superiority of any (or, perhaps, any differences were confounded with changes in

the other operational parameters). For general use, extended arithmetical crossover

appears a safe choice, as it provides some searching plus good mixing of the parents

genetic material. The balance between mutation and recombination remains impor-

tant, and using approximately equal weightings of these two key operators generally

ensures their synergistic effect.

6. Beef property optimisations

Despite the promising results of the real-value genetic algorithm in Table 1, we

still require the most efficient implementation of this method, as only a limit of

around 0.1 million model runs is computationally feasible. From the rates of con-

vergence of the evolutionary algorithm optimisations of the 40-dimensional herd

dynamics model (not presented), the two outstanding performers were the optimi-

sations using higher population numbers (100 and 500, rather than 50), combined

with moderate-level double mutation operators.

Time limitations only allowed four optimisations of the whole-property model. All

used deterministic tournament selection of parents (with a tournament size of two),

as this method is computationally more efficient than ranking and/or weighting the

populations scores each iteration. Sub-generational replacement of the worst indi-

viduals was also used, and this guarantees elitism. The balance between crossoverand mutation, as controlled by weighting factors in GENIAL, was set at a ratio of 1:1.

Following considerations of all previous results, the four optimisation methods

chosen consisted of a factorial of two population sizes (namely 200 with 150 of these

being replaced each generation, and 500 with 400 replacements), by two clusters of

operations. The lower population size (200) was selected as being between two and

three times the dimensionality of the beef model, and 500 was chosen as a larger

comparative level to ensure adequate genetic diversity is maintained in the popula-

tion. The first of the clusters of operations represents a more standard group of

operators, namely uniform crossover (with a crossover probability of 0.50, and

weight of 50%) combined with double mutation (low-level Gaussian creep mutation

at a probability of 0.05 and weight of 25%, plus moderate-level boundary muta-tion of probability 0.20 and weight 25%).

The second cluster of operators is more comprehensive, and was selected to allow

a shotgun approach in the absence of much knowledge of the relative effective-

ness of various parameters (given their potentially different weights and prob-

abilities), a large number of recombination and mutation operators can be applied

to the problem. The expected advantage of this is that different operators will prove

more effective during the various stages of the optimisation. For example, expansive

operators will be particularly useful early (during exploration), and operators which


8/14/2019 1-s2.0-S0308521X01000257-main

11/15

introduce smaller-scale changes will be better in the latter (fine-tuning) phases. This

shotgun cluster consisted of four crossover and 12 mutation operators. The cross-

overs were one-point (weight of 5%), two-point (same weight), uniform (weight of20% and probability of 0.50), and random extended arithmetical crossover (weight

of 22% and probability of 0.50). This gives a total crossover weighting of 52%,

leaving 48% for the remaining mutation operators, and thus approximately main-

taining the 1:1 ratio between the operator types. The 12 mutation operators of this

shotgun approach were all given a weight of 4%, and ranged widely in terms of

types and probabilities, as listed in Table 3. The minimum probability of 0.02 was

chosen to give an expectation of about one gene mutated per new population member.

Each of the four investigative optimisations was terminated either manually or as

a result of power failures, which occurred with annoying regularity during the sum-

mer months. These optimisations generated between 25 300 and 63 300 model runs

each, with the longest taking 39 days (and, incidentally, running right across the

threatened Y2K bug). As expected, and evident in Fig. 1, these optimisations had

not converged adequately within these numbers of model runs. However, they were

at least approaching their asymptotic phases.

The final management options thus identified by the four optimisations varied

somewhat, and probably few are exactly optimal, as the routines had not converged.

As expected, the standard operator optimisations (which had boundary mutation

at a weighting of 25%, compared to a total of only 16% under the shotgun

approach) had more of their options on the boundaries. This appears to have been

effective on this particular problem, because for each population size the standard

operators resulted in economic superiority (Fig. 1). The shotgun operators would

(presumably) have caught up eventually, as the other forms of mutation crept onto

these management boundaries, and perhaps then their more comprehensive searchpattern may have found better solutions. However, within available computation,

this possibility could not be investigated.

The lower population size of 200 appears superior, especially when used with the

standard operators. This shows that a population of around three times the prob-

lems dimensionality (of 70) did contain sufficient genetic diversity, whereas carrying

larger numbers (500) tended to slow down the overall process. One further trial

optimisation (to only 104 model runs), using the standard operators and a popula-

tion of 150 (being approximately twice the dimensionality), gave very similar results

to the optimisation using 200. This suggests that about two-plus is an appropriate

Table 3

Mutation operators used in the shotgun genetic algorithm

Type of mutation Mutation probabilities

0.02 0.05 0.10 0.20 0.50

Uniform @ @ @

Uniform creep @ @

Gaussian creep @ @ @

Boundary @ @ @ @


8/14/2019 1-s2.0-S0308521X01000257-main

12/15

multiplier of the problems dimensionality to use in practice. It would appear

imprudent to go much below this, especially with new problems.

Finally, to test the possible effect of the balance of crossover to mutation opera-

tors, two further replicate optimisations were also conducted. These used the best

combination from Fig. 1 (a population size of 200 and the standard set of opera-

tors), but had a crossover to mutation ratio of 1:2. Unfortunately, both replicates

were terminated fairly early by power failures, at 8400 and 5500 model runs respec-

tively. Up until these points, their rates of convergence straddled the solid line of

Fig. 1 (one replicate was continually above, and the other below). This indicates,once again, that the exact balance between these primary operators of evolutionary

algorithms is not critical.

7. Conclusions

For the efficient optimisation of large, complex models of agricultural systems,

evolutionary algorithms have proven to be the most efficient and robust of the

Fig. 1. Rates of convergence for evolutionary algorithm optimisations of beef property model (solid line

for standard operators and population size of 200; short-dash for standard operators and 500; long-dash for shotgun operators and 200; dot-dash for shotgun operators and 500).


8/14/2019 1-s2.0-S0308521X01000257-main

13/15

currently-available optimisation methods. Suggested operational settings which have

been shown to work well on this series of agricultural models include:

1. real-value coding;

2. a population size of the order of two times the dimensionality of the problem;

3. tournament selection of the parents, with a tournament size of two;

4. steady-state deterministic replacement of the worst individuals (thus assuring

elitism);

5. at least two different concurrent mutation operators, each with a probability in

the range of 0.050.20;

6. recombination via extended arithmetical crossover; and

7. the total of the mutation operators having approximately equal weighting to

that of the recombination operators.

Acknowledgements

We are grateful to Lester Ingber, John Grefenstette and Henrik Widell for making

their optimisation algorithms available for general use.

References

Annevelink, E., 1992. Operational planning in horticulture: optimal space allocation in pot-plant nur-

series using heuristic techniques. Journal of Agricultural Engineering Research 51, 167177.

Ba ck, T., 1997. Mutation parameters. In: Ba ck, T., Fogel, D.B., Michalewicz, Z. (Eds.), Handbook of

Evolutionary Computation. Oxford University Press, New York, pp. E1.2:1E1.2:7.

Ba ck, T., Schwefel, H.-P., 1993. An overview of evolutionary algorithms for parameter optimization.

Evolutionary Computation 1, 123.

Barioni, L.G., Dake, C.K.G., Parker, W.J., 1997. Optimising rotational grazing in sheep management

systems. Proceedings 1997 International Congress on Modelling and Simulation, 811 December 1997,

University of Tasmania, Hobart, pp. 10681073.

Blickle, T., Thiele, L., 1997. A comparison of selection schemes used in evolutionary algorithms. Evolu-

tionary Computation 4, 361394.

Bos, J., 1993. Zoning in forest management: a quadratic assignment problem solved by simulated

annealing. Journal of Environmental Management 37, 127145.

Botes, J.H.F., Bosch, D.J., Oosthuizen, L.K., 1996. A simulation and optimization approach for evalu-

ating irrigation information. Agricultural Systems 51, 165183.

Bratley, P., Fox, B.L., Schrage, L.E., 1987. A Guide to Simulation. Springer-Verlag, New York.

Bunday, B.D., 1984. Basic Optimisation Methods. Edward Arnold, London.Corana, A., Marchesi, M., Martini, C., Ridella, S., 1987. Minimizing multimodal functions of continuous

variables with the simulated annealing algorithm. ACM Transactions on Mathematical Software 13,

262280.

De Jong, K., Sarma, J., 1995. On decentralizing selection algorithms. Proceedings Sixth International

Conference on Genetic Algorithms, 1519 July 1995, University of Pittsburgh, pp. 1723.

Dent, J.B., Blackie, M.J., 1979. Systems Simulation in Agriculture. Applied Science Publishers, London.

Fletcher, R., 1987. Practical Methods of Optimization. Wiley, New York.

Fogel, D.B., 1995. Evolutionary Computation Toward a New Philosophy of Machine Intelligence.

IEEE Press, New York.


8/14/2019 1-s2.0-S0308521X01000257-main

14/15

Glover, F., Taillard, E., de Werra, D., 1993. A users guide to tabu search. Annals of Operations Research

41, 328.

Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization and Machine Learning. Addison-

Wesley, Reading.

Hammel, U., 1997. Simulation models. In: Ba ck, T., Fogel, D.B., Michalewicz, Z. (Eds.), Handbook of

Evolutionary Computation. Oxford University Press, New York, pp. F1.8:1F1.8:9.

Hart, R.P.S., Larcombe, M.T., Sherlock, R.A., Smith, L.A., 1998. Optimisation techniques for a com-

puter simulation of a pastoral dairy farm. Computers and Electronics in Agriculture 19, 129153.

Hendrickson, J.D., Sorooshian, S., Brazil, L.E., 1988. Comparison of Newton-type and direct search

algorithms for calibration of conceptual rainfall-runoff models. Water Resources Research 24, 691700.

Hinterding, R., Gielewski, H., Peachey, T.C., 1995. The nature of mutation in genetic algorithms. Pro-

ceedings Sixth International Conference on Genetic Algorithms, 1519 July 1995, University of Pitts-

burgh, pp. 6572.

Holmes, W.E., 1995. BREEDCOW and DYNAMA Herd Budgeting Software Package. Queensland

Department of Primary Industries, Townsville.

Horton, B., 1996. A method of using a genetic algorithm to examine the optimum structure of the Aus-

tralian sheep breeding industry: open-breeding systems, MOET and AI. Australian Journal of Experi-mental Agriculture 36, 249258.

Ingber, L., 1993. Simulated annealing: practice versus theory. Mathematical and Computer Modelling 18,

2957.

Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., 1983. Optimization by simulated annealing. Science 220,

671680.

Kuo, C.-H., Michel, A.N., Gray, W.G., 1992. Design of optimal pump-and-treat strategies for con-

taminated groundwater remediation using the simulated annealing algorithm. Advances in Water

Resources 15, 95105.

Lagarias, J.C., Reeds, J.A., Wright, M.H., Wright, P.E., 1998. Convergence properties of the Nelder-

Mead simplex algorithm in low dimensions. SIAM Journal on Optimization 9, 112147.

Lockwood, C., Moore, T., 1993. Harvest scheduling with spatial constraints: a simulated annealing

approach. Canadian Journal of Forestry Research 23, 468478.

Mayer, D.G., Belward, J.A., Burrage, K., 1996. Use of advanced techniques to optimize a multi-

dimensional dairy model. Agricultural Systems 50, 239253.Mayer, D.G., Belward, J.A., Burrage, K., 1998a. Optimizing simulation models of agricultural systems.

Annals of Operations Research 82, 219231.

Mayer, D.G., Belward, J.A., Burrage, K., 1998b. Tabu search not an optimal choice for models of agri-

cultural systems. Agricultural Systems 58, 243251.

Mayer, D.G., Belward, J.A., Burrage, K., 1999a. Performance of genetic algorithms and simulated

annealing in the economic optimization of a herd dynamics model. Environment International 25,

899905.

Mayer, D.G., Belward, J.A., Burrage, K., Stuart, M.A., 1995. Optimisation of a dairy farm model com-

parison of simulated annealing, simulated quenching and genetic algorithms. Proceedings 1995 Interna-

tional Congresson Modellingand Simulation, 2730 November 1995, University of Newcastle,pp. 3338.

Mayer, D.G., Belward, J.A., Widell, H., Burrage, K., 1999b. Survival of the fittest genetic algorithms

versus evolution strategies in the optimization of systems models. Agricultural Systems 60, 113122.

Mayer, D.G., Pepper, P.M., McKeon, G.M., Moore, A.D., 1999c. Modelling mortality and reproductionrates for management of beef herds in northern Australia. Proceedings Applied Modelling and Simu-

lation International Conference, International Association of Science and Technology for Develop-

ment (IASTED), 13 September 1999, Cairns, pp. 142147.

Mayer, D.G., Schoorl, D., Butler, D.G., Kelly, A.M., 1991. Efficiency and fractal behaviour of opti-

misation methods on multiple-optima surfaces. Agricultural Systems 36, 315328.

Michalewicz, Z., 1996. Genetic Algorithms+Data Structures=Evolution Programs, 3rd Edition.

Springer, Berlin.

Mu hlenbein, H., Schlierkamp-Voosen, D., 1994. The science of breeding and its application to the breeder

genetic algorithm BGA. Evolutionary Computation 1, 335360.


8/14/2019 1-s2.0-S0308521X01000257-main

15/15

Nelder, J.A., Mead, R., 1965. A simplex method for function minimisation. The Computer Journal 7,

308313.

ORourke, P.K., Winks, L., Kelly, A.M., 1992. North Australia beef producer survey 1990. Queensland

Department of Primary Industries, Brisbane.

Parsons, D.J., 1998. Optimising silage harvesting plans in a grass and grazing simulation using the revised

simplex method and a genetic algorithm. Agricultural Systems 56, 2944.

Paz, J.O., Batchelor, W.D., Babcock, B.A., Colvin, T.S., Logsdon, S.D., Kaspar, T.C., Karlen, D.L., 1999.

Model-based techniqueto determine variable rate nitrogen for corn. Agricultural Systems 61, 6975.

Peck, C.C., Dhawan, A.P., 1995. Genetic algorithms as global random search methods: an alternative

perspective. Evolutionary Computation 3, 3980.

Roise, J.P., 1990. Multicriteria nonlinear programming for optimal spatial allocation of stands. Forest

Science 36, 487501.

van Laarhoven, P.J.M., Aarts, E.H.L., 1987. Simulated Annealing: Theory and Applications. Reidel,

Dordrecht.

Wang, Q.J., 1991. The genetic algorithm and its application to calibrating conceptual rainfall-runoff

models. Water Resources Research 27, 24672471.

Watson, R.A., Sumner, N.R., 1997. Maximising the landed value from prawn fisheries using a variationon the simulated annealing algorithm. Proceedings 1997 International Congress on Modelling and

Simulation, 811 December 1997, University of Tasmania, Hobart, pp. 864868.

Widell, H., 1997. GENIAL 1.1. A function optimizer based on evolutionary algorithms user manual.

Available at: http://www.hjem.get2net.dk/widell/genial.htm.


1-s2.0-S0308521X01000257-main

Documents

Transcript of 1-s2.0-S0308521X01000257-main