Optimising the management of maize – Grevillea robusta fields in Kenya

Mbae N. Muchiri1, Timo Pukkala2,* and Jari Miina2

1Kenya Forestry Research Institute, Box 20412, Nairobi, Kenya; 2Faculty of Forestry, University of Joensuu,Box 111, FIN-80101 Joensuu, Finland; *Author for correspondence (e-mail: timo.pukkala@forest.joensuu.fi;phone: +358 13 2514092; fax: +358 13 2514444)

Received 29 November 2000; accepted in revised form 22 October 2001

Key words: Competition index, Penalty function, Silky oak, Simulation model, Uneven-aged management, Zeamays

Abstract

The study optimised the management of a Grevillea robusta (A. Cunn.) stand growing in the central highlands ofKenya. The optimisations were conducted separately for even-aged and uneven-aged management system oftrees. The management was also optimised with the requirement that maize production under the tree cover mustbe profitable every year. Technically, the optimisation problems were solved by linking a simulation programwith the non-linear optimisation algorithm of Hooke and Jeeves. The simulation program calculated the treegrowth, volumes of harvested trees, and maize yields with a given set of management parameters (decision var-iables). The maize yield predictions and simulated timber yields were converted into gross incomes of which theproduction costs were subtracted. In even-aged management the objective variable was the soil expectation valuewith 5% discounting rate. In uneven-aged management the mean annual net income was maximised. The optimalsolutions indicated that with both management systems it is optimal to concentrate on wood production. Theoptimal stand densities were so high that profitable maize production was not possible under the tree cover. Themean annual wood production of the optimal management schedule was more than 50 m3 ha−1. Forcing profit-able maize production in the solution decreased the wood production by 57% (even-aged forestry) or 27% (un-even-aged forestry) and net income by 45% (even-aged forestry) or 24% (uneven-aged forestry).

Introduction

Maize (Zea mays L.) is the staple food for small-scalefarmers and their families in the highlands of CentralKenya. The maize – G. robusta agroforestry systemcovers about 750 000 ha in the highland area aroundMt Kenya. The majority of farmers usually growmaize for their own consumption. Nonetheless, someof it is sold to raise money to buy foods rich in pro-teins and vitamins, pay school fees of children andprovide other basic human needs like clothing. Occa-sionally, a family may sell maize to raise money foruncertainties (e.g. sudden hospitalisation).

G. robusta is the dominant component of the treevegetation cover in the maize – G. robusta agrofor-estry system in Kenya and is usually grown to pro-duce timber, poles and firewood for sale. Yet, mostfamilies are entirely dependent on G. robusta fire-

wood for cooking and warming. Because the trees arenot felled before they attain a size that can producetimber, about 80 percent of the firewood is harvestedwhen G. robusta is pruned and pollarded, and the re-mainder when suppressed trees are removed and largetrees are harvested. G. robusta is planted either inrows (alley cropping) or in less regular spatial ar-rangements (intercropping). G. robusta is inter-cropped with maize, beans, bananas, coffee, etc.Sometimes G. robusta is planted in pure stands(wood-lots) but the most common practice is to plantalong the boundaries of a land parcel belonging to onefamily. The latter practice has resulted in conflictsamong farmers because a dense row of large trees re-duces the crops yield and tree growth of the neigh-bour farmer.

The income from maize is realised three monthsafter the maize is planted and once every year (some-

13Agroforestry Systems 56: 13–25, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

times every six months) while that from G. robusta isrealised five or more years after planting. Because ofthis reason and the ever increasing pressure on thefarmland due to rapid increase in population (the an-nual rate of population growth was about 3.0 percentin 1989–1999), farmers have cultivated almost all thefarmland with food crops. G. robusta and other treesare being cleared on farms because they compete withfood crops and pastures. The farmers are however indire need for firewood and construction material. Theconcern is, are the farmers maximising the economicbenefit from their land units because their primaryland use is maize production, which may not take intoaccount the opportunity cost of producing only maize.

The objective of this study was to find the optimalmanagement of G. robusta stands with and withoutprofitable maize intercropping being necessary in thesame area. The management was optimised separatelyfor even- and uneven-aged management systems.Even-aged management suits to woodlots whereasuneven-aged management is a more relevant systemin agroforestry fields. However, maize production isnot excluded from even-aged stands, either. The op-timal management was derived separately, besideseven- and uneven-aged forest management, also forsituations where maize production must be profitableevery year and situations in which it does not matterwhether profitable maize production is possible ornot.

Methods

The simulation – optimisation system

Optimising the management of G. robusta is equal tofinding the best combination of a set of managementparameters, which in this context are called decisionvariables. The best combination is the one that maxi-mises the objective function. In even-aged manage-ment the objective function of this study was the soilexpectation value (SEV), which is the net presentvalue of the incomes and costs of the current rotationand all future similar rotations. In uneven-aged man-agement, in which the idea is to keep the stand struc-ture unchanged by cutting trees gradually at short in-tervals, the objective function was the mean annualincome.

In even-aged management the first decision varia-ble is the number of seedlings to be planted in anopen area, and the last decision variable is the clear-

felling year. Management also includes decisionsabout thinnings. In this study, the whole managementsystem was specified using the following decisionvariables:

1. Number of planted trees per hectare (N)2. Between-tree competition level which induces a

thinning treatment (Thin-CItree)3. Thinning percentage (Thin-%)4. Rotation length (R)

Thin-CItree was measured by a mean competitionindex of trees. The competition index of a tree wascalculated using the same formula as in the distancedependent diameter model of Muchiri et al. (2001a)(in press) for G. robusta:

CItree � �k � 1

n

dk2/sk (1)

where dk is diameter (cm) and sk is distance (m) ofcompetitor k and n is the number of competitors. Allneighbour trees nearer than 8 metres are included inthe competition index.

In uneven-aged forestry the set of decision varia-bles must be specified in a different way. Plantingdensity and rotation length are not required to specifythe management regime. The management consists offinding a proper diameter (and age) distribution andselection thinning method for the stand. In an eco-nomically optimal management the diameter distribu-tion and stand density are selected so that the yield ismaximised, and the selection thinning is conducted sothat the stand structure always reaches the same pre-thinning state just prior to next thinning. If there isno natural regeneration, it is possible to plant newtrees after the thinning or any other time. The timeinterval between two successive cuttings in the samestand is called cutting cycle.

In our study, the diameter distribution of the standin the beginning of a cutting cycle was described withthe Weibull function (e.g. Bailey and Dell (1973) andGove and Fairweather (1989), Kilkki et al. (1989),Maltamo et al. (1995)). The probability density func-tion of the three-parameter Weibull distribution for arandom variable d (tree diameter) is:

f�d� �c

b�d � a

b �c � 1exp�� �d � a

b �c�, a � d � �

(2)

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where d is tree diameter at the breast height (cm); anda, b and c are parameters which define the location(minimum), scale (range) and shape (skewness) of thedistribution, respectively. We fixed the minimum di-ameter (parameter a) to one cm, which is close to thedbh of a large seedling.

When the stand structure is described with theWeibull distribution, and its parameter a is fixed, thewhole set decision variables in uneven-aged manage-ment becomes as follows:

1. Number of trees per hectare of the stand prior toselection thinning (N)

2. Parameter b of the Weibull distribution prior toselection thinning (b)

3. Parameter c of the Weibull distribution prior to se-lection thinning (c)

4. Number of trees harvested from different diameterclasses (H1, H2, H3)

5. Number of new trees planted after the selectionthinning (Nnew)

The harvested number of trees was specified sepa-rately for the following three diameter classes: 5–9.99cm, 10–19.99 cm, and 20 cm or more. These corre-spond to different timber assortment and unit pricecategories. This implies that trees smaller than 5 cmat the breast height are never cut. Because the harvestpercentages were separate for different diameterclasses, the total number of decision variables thatwas needed for specifying the management systemwas seven.

The optimal management, i.e. the optimal combi-nation of decision variables, was found with a com-bined use of iterative optimisation algorithm (Hookeand Jeeves 1961) and a simulation model. A givencombination of decision variables (DVs) is fed intothe simulator, which simulates the temporal develop-ment of the stand, simulates cuttings, calculates themaize yields, and from these results calculates thevalue of the objective function. The objective func-tion value is passed back to the optimisation algo-rithm, which makes alterations in the values of deci-sion variables, based on the feed-back from thesimulation program (Figure 1). This process is re-peated many times, until a user-specified stopping cri-terion is met. The final combination of decision var-iables is the optimal one, or close to it, depending onthe stopping criterion and optimisation algorithm.

Simulating the development of G. robusta stand

The temporal development of G. robusta trees wassimulated with the same models in both even-agedand uneven-aged management systems. In even-agedforestry, the simulation proceeded as follows:

1. Generate the initial stand (a new plantation);2. Simulate the growth and mortality of trees until

the next cutting;3. Simulate the cutting; and4. Repeat steps 2 and 3 until the rotation length is

reached.

In uneven-aged forestry the simulation steps wereslightly different:

1. Generate an uneven-aged prior-thinning stand (ini-tial stand);

2. Simulate a partial cutting;3. Plant new trees; and4. Simulate tree growth and mortality to the end of

the cutting cycle.

The initial stand description consisted of a list oftrees in a rectangular plot, each tree being describedby the following variables: diameter, height, age, andx and y coordinates. In even-aged forestry, where sim-ulation began with an open area, the only decisionvariable affecting the initial stand was the number ofplanted trees ha−1. This number was multiplied by0.85 because a 15% initial mortality was assumed(Wanyiri et al. 2000). The remaining seedlings weregiven a dbh of 1 cm, height of 1.5 m, and age of oneyear. The trees were Poisson-distributed over thefield, i.e. the x and y coordinates were randomly

Figure 1. The principle of using a simulation program and an op-timisation algorithm for optimising the management of Grevillearobusta – maize agroforestry system.

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drawn from uniform distribution. The Poisson distri-bution is the easiest to simulate and it correspondsfairly well to actual intercropping fields.

In uneven-aged forestry the tree diameters of theinitial stand were drawn from the Weibull distributionspecified by the scale and shape parameters (b and cin Equation (2)). First, a random diameter was gen-erated between 1 and 35 cm. Then its frequency wascalculated using Equation (2). Then a random num-ber was drawn from a uniform distribution and com-pared to this frequency. If the random number wassmaller than the frequency (Equation 2) the tree can-didate was accepted, i.e. the tree was included in thetree list. The tree coordinates were generated in thesame way as in the even-aged case. The age andheight of the trees were calculated using (Muchiri etal. 2001a (in press)):

t � 0.560d � 0.0346d�P (3)

ln�h� � 3.683 � 1067.326/��t � 30��d � 10�� (4)

where t is tree age (a); d is dbh (cm); h is tree height(m); and P = 1 if the tree has been pollarded and 0otherwise. In this study, pollarding was never done,i.e. P was always zero.

With both systems, tree growth and mortality weresimulated as follows:

1. Increment tree ages by one year;2. Compute breast height diameters (dbh) corre-

sponding to the new age by using a non-spatial di-ameter model (Equation 5). Take the effect ofcompetition into account by re-computing all di-ameters using a spatial diameter model (Equation6), and repeat the re-computing until the diametersconverge (five times in this study);

3. Compute tree heights corresponding to the newage and diameter using a non-spatial height model(Equation 4);

4. Specify trees that belong to the annual mortality,and remove these trees from the tree list;

5. Calculate tree volumes and stand characteristics.

The following models by Muchiri et al. (2001a) (inpress) were used to calculate tree diameter:

Non-spatial diameter model:

ln�d� � 4.668 � 32.749/�t � 8� (5)

Spatial diameter model:

ln�d� � 4.744 � 33.126/�t � 8� � 0.398ln�CItree � 1�(6)

where d is dbh (cm), t is tree age (a), and CItree iscompetition index computed using Equation (1).Whenever competition indices were computed, a10-m buffer zone was temporarily added to the plotby assuming that similar plots surrounded the plot onall sides. The volume function developed by Mabvur-ira and Eerikäinen (2001) (in press) for Eucalyptusgrandis was used to calculate stem volumes (v, dm3)because a volume function of G. robusta does not ex-ist.

ln�v� � � 3.872 � 0.389ln�d� � 2.681ln�h�

� 1.350ln�d/�h � 1.3�� (7)

The study by Muchiri et al. (2001a) (in press) doesnot include a mortality model. Although tree mortal-ity may be negligible in sparsely populated agrofor-estry fields, and may be omitted, the situation is dif-ferent in optimisations where many different standdensities are tried. In the absence of a mortalitymodel, the simulation of tree mortality was based onthe competition index (Equation 1) (e.g. Keister(1972) and Monserud (1976)). In the study materialof Muchiri et al. (2001a) (in press) the distribution ofthe competition index of Equation (1) showed thatCItree was higher than 0.5 only for a few trees andthe main part of the distribution ended at CItree = 0.5(Figure 2). It was concluded that those few trees forwhich CItree was higher than 0.5 were exceptionallypersistent individuals, or they were surrounded bytree species other than G. robusta. Based on this ra-tionale, all trees for which the CItree passed the 0.5-limit during one-year growth period were regarded asmortality.

Calculation of maize yield

Muchiri et al. (2001b) developed a random parametermodel for describing the effect of G. robusta trees onthe biomass of maize in the maize – G. robusta agro-forestry fields. This model consists of a fixed part anda random part. The random part accounts for the dif-ferences between fields due to soil fertility, fertilisa-tion, genetic quality of the maize seed, etc. In thisstudy, the model of Muchiri et al. (2001b) was re-placed by a new model for two reasons. The first rea-

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son is that the model does not explicitly describe theeffect of fertilisation and the use of genetically im-proved seed on the maize yield because these effectsare embedded in the random field factors. However,in economic calculations it is necessary to know, forestimating the production costs correctly, which typeof seed is used and whether the maize plants are fer-tilised.

The other reason for replacing the model by a newone is that although the model form gave the best fitwithin the study material of Muchiri et al. (2001b), itmay not be the best form with higher stand densitiesencountered in optimisation studies. According to themodel of Muchiri et al. (2001b), the effect of compe-tition by trees on maize yields almost ceases to in-crease at certain competition levels. However, itwould be more logical to assume that the maize yieldcontinues to decrease as a function of increased com-petition until the yield is very close to zero.

The new model for maize biomass, developed us-ing the same material as (Muchiri et al. 2001b), is asfollows (Figure 3):

Biomass � 0.327 � 0.188FERT � 0.119IMP

� 0.166CImaize (8)

where Biomass is the fresh above-ground biomass ofmaize plants (kg m−2), CImaize is competition indexcalculated from the point-to-tree distances and treediameters, FERT = 1 if the field is fertilised, other-wise FERT = 0, and IMP = 1 if the field is fertilisedand planted with genetically improved maize seed,otherwise IMP = 0.

The competition index CImaize of Equation (8) isas follows (Muchiri et al. 2001b):

CImaize � �k � 1

n

dk/sk (9)

where dk is diameter (cm) and sk is distance (m) ofcompetitor k and n is the number of competitors. Alltrees closer than 10 metres to the calculation point areincluded in the competition index.

To be able to calculate the economic return frommaize cultivation, another model was developed forpredicting the share of grain of the total maize bio-mass:

Grain � 0.167 � Biomass2/�381.5 � 2.077Biomass�2

(10)

This function was estimated from the same data asused in the study by Mwihomeke et al. (1999). Thefunction was used to calculate the grain yield ofmaize in a given year. The maize yield of the plot wasestimated by predicting the yield at 5-m intervals in xand y directions using Equations (9) and (10), andmultiplying the sum of these predictions by the arearepresented by one calculation point (5 m × 5 m = 25m2).

Calculation of objective function value

In even-aged forestry the objective function was thesoil expectation value which is equal to the net

Figure 2. Distribution of between-tree competition index CItree

(Equation 1) in the study material of Muchiri et al. (2001a) (inpress) in Kenya. Figure 3. Dependence of above-ground maize biomass on the

competition by trees (CImaize, Equation (9)) according to the model(lines) and according to tentiles calculated for different sub-sets ofmodelling data of Muchiri et al. (2001b) in Kenya. FERT meansthat the maize field is fertilised and IMP means that geneticallyimproved maize seed has been used.

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present value of all future incomes. It included thefollowing components: establishment costs of the G.robusta stand, incomes from timber sales, maize pro-duction costs, and income from maize. The stand es-tablishment cost (Cw, USD ha−1) was calculatedfrom:

Cw � 55 � 0.352N (11)

where N is number of planted trees ha−1. The func-tion tells that there is an area-dependent cost of 55USD ha−1 plus a cost of 0.352 USD per seedling(seedling price including transport, planting work,etc). The income from timber sales (Iw, USD ha−1)was computed from:

Iw � �PiHi � �5 � 2�Hi� (12)

where Pi is unit roadside price (USD m−3) in diam-eter class i, and Hi is volume harvested from diam-eter class i. The unit price was 7.5 USD m−3 in di-ameter class 5–9.99 cm, 17.1 USD m−3 in diameterclass 10–19.99 cm, and 35.7 USD m−3 in diameterclass 20 cm or more. The fixed harvesting cost was 5USD ha−1 and the unit cost was 2 USD m−3.

USD 272 ha−1 was used as the maize productioncost (Cm). This cost estimate is based on the use ofgenetically improved maize seed and fertilisation ofthe maize plants. Accordingly, the dummy variablesFERT and IMP of Equation (8) were set equal to onewhen predicting maize yields. The income frommaize (Im, USD ha−1) was calculated from:

Im � 0.2GRAIN � 0.01RESIDUES (13)

where GRAIN is grain production in kg ha−1, andRESIDUES is the fresh mass of the other above-ground biomass components in kg ha−1 (RESIDUES= BIOMASS − GRAIN).

The price of wood in diameter class more than 20cm was as given by Kenya Forestry Department Gen-eral Order No. 250 (Ministry of Environment andNatural Resources 2000). For wood in diameter classless than 20 cm it was the price of firewood and polesat KEFRI, Muguga in year 2000. The price of theseedling was that at KEFRI, Muguga in year 2000plus the cost of the farmer’s time spent on plantingthe seedling and transport cost within a radius of 5km. The price of maize grain and maize crop residualwas the year 2000 market price in the study area. Themaize production costs include the cost of improved

seed, cost of fertiliser, hand tools, and variable costsassociated with growing maize, harvesting, packagingand looking for market.

A discounting rate of five percent was used in allcalculations. Stand establishment costs were not dis-counted because the stand was established in the be-ginning of the first year. Maize production costs werediscounted from the beginning of each year, and in-come from grain and crop residues 0.33 years later(the maize production time is four months). However,if the net present value of maize production was notprofitable in a given year (C m > I m/1.050.33) it wasassumed that there is no maize production at all, andboth C m and I m were set to zero. Net incomes fromcuttings were discounted from the cutting year.

The simulation always covered one full rotation.The later rotations were assumed to be replicates ofthe first rotation, which means that the net presentvalue of the other rotations was obtained by multiply-ing the net present value of the first rotation by1/(1.05 R − 1) where R is rotation length in years.

In uneven-aged forestry the calculations werefairly similar except that the net incomes were notdiscounted and there was no stand establishment costsince the simulation began with an older stand. Be-cause the idea of uneven-aged forestry is to keep thestand structure unchanged, a penalty function wasadded to the objective function (e.g. Bazaraa and Sh-etty (1979)). The value of the penalty increased as afunction of the difference between the initial and thefinal stand characteristics. The penalty function wasas follows:

Penalty � p1¦V �end� � V �ini�¦p2 � p3�¦N �end�i � N �ini�i¦

p2

(14)

where V (end) and V (ini) are stand volumes in the be-ginning and at the end of simulation, respectively;and N (end)i and N (ini)i are, respectively, the final andinitial frequency of diameter class i. The diameterclasses used in penalty function were: 1–1.99 cm,2–4.99 cm, 5–9.99 cm, 10–19.99 cm, and 20 cm ormore. Many different values of penalty parametersp1, p2 and p3 were tried, and the following valueswere finally used: p1 = 10, p2 = 1 and p3 = 1. Theobjective function (OF), which was maximised whenoptimising the management of uneven-aged G. ro-busta stand, was

OF � Mean annual net income � Penalty (15)

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It is possible to make the optimisation problem astrictly constrained one by setting the penalty param-eters large enough. However, it was noticed that verylarge penalty parameters lead to solutions that are farfrom optimal (produce low incomes) as the directsearch procedure becomes too constrained (Wikströmand Eriksson 2000).

Optimisation procedure

Optimising the management of a maize-Grevillea ro-busta field using the simulation model described re-quires the use of a non-linear optimisation algorithmbecause the model is not convex and differentiable inthe decision variables, the model has constrains re-quiring penalty function, and the response surfacesare not smooth. Therefore, the simulator was linkedwith the direct search method of Hooke and Jeeves(1961), an algorithm commonly used for optimisingthe management of both even-aged stands (e.g. Roise(1986) and Valsta (1990), Miina (1996), Rautiainen etal. (1999)) and uneven-aged stands (e.g. Bare andOpalach (1987) and Haight and Monserud (1990)). Asthe name implies, direct search method uses only ob-jective function values, retaining a certain number ofcombinations of decision variables which are itera-tively improved. The direct search starts from an ini-tial point. The search for a better solution is done bothalong the decision variables to discover the pattern ofthe objective function, and along the discovered pat-tern direction (for more details see e.g. Bazaraa andShetty (1979)). The search step is reduced if thesearch does not provide a better solution.

Because the Hooke and Jeeves method is sensitiveand inconsistent with respect to the number of deci-sion variables, starting points, the feasible range ofdecision variables, etc., it is common to solve theproblem for several starting points and compare thesesolutions. In this study, each optimisation problemwas solved for eleven different initial vectors of de-cision variables. Each of these was chosen as the bestof 200 random search trials. The whole system of 11direct search runs was repeated 2–3 times, first withthe widest possible feasible range of each decisionvariable to find an approximate optimum, and thenwith somewhat narrower ranges of decision variables.In even-aged stand management, the search for bettermanagement regimes was stopped when the step sizesfor changing the number of planted trees per hectare(N), between-tree competition for inducing a thinning(Thin-CItree), thinning percentage (Thin-%), and rota-

tion length (R) were 5 ha−1, 0.0003, 0.1% and 0.05years, respectively. In uneven-aged stand manage-ment the corresponding values for stand density (N),Weibull parameters (b and c), the number of treesharvested from different diameter classes (H1, H2,H3), and the number of new trees planted after thecutting (Nnew) were 3 ha−1, 0.018, 0.0018, 0.050ha−1, 0.1 ha−1, 0.6 ha−1 and 1 ha−1, respectively.

Forcing profitable maize production in the solution

The purpose of the study was to find out an overalloptimum for even-aged and uneven-aged manage-ment system, without profitable maize production be-ing necessary every year. The other objective was tofind such optima, which maximise the objective func-tion with the constraint that maize production must beprofitable in every year. In even-aged forestry, theconstrained ‘optimum’ was searched by using simpletrial and error: with a given planting density, rotationlength and thinning percentage, the pre-thinning com-petition level (Thin-CItree) was lowered until the standdensity was continuously so low that maize produc-tion was profitable throughout the rotation. In uneven-aged forestry an additional penalty of 1000 USD ha−1

was subtracted from the objective function in everyyear in which maize production was not profitable.

Results

Even-aged management

Even-aged management without requirement ofprofitable maize productionThe optimal solution for even-aged management wasas follows: planting density (N) 6475 trees ha−1; meanbetween-tree competition index activating a thinningtreatment (Thin-CItree) 0.29; thinning percentage(Thin-%) 14%; and rotation length (R) 20 years (Ta-ble 1). Thinnings begin 7 years after planning afterwhich the 14-% thinning is repeated every year (Fig-ure 4). The value of the objective function (SEV) forthis schedule is 20776 USD ha−1 and growing maizeis profitable during the first year only. The mean an-nual wood harvest is 56 m3 ha−1, of which 36 m3 ha−1

are from thinnings and 20 m3 ha−1 from clear-felling.The net annual income is 1547 USD ha−1.

The sensitivity of the objective function value(SEV) to changes in the values of the DVs was testedby increasing or reducing the value of one DV at a

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time by 10, 20 or 30 percent and resimulating themanagement schedule. The SEV was not sensitive tosmall changes in the DVs (Table 2). SEV was how-ever rather sensitive to Thin-CItree levels that aremuch lower than the optimal Thin-CItree. SEV is alsorather sensitive to rotations that are much shorter orlonger than the optimal rotation. The sensitivity anal-yses of Table 2 and the DV values of the best solu-tions (Table 1) indicate that it is sufficient to specifythe optimal management schedule as follows: plant-ing density (N) 5000–7000 trees ha−1; prior-thinningcompetition level (Thin-CItree) 0.25–0.35; thinningpercentage (Thin-%) 12–15%; and rotation length (R)19–21 years.

Even-aged management with a requirement ofprofitable maize productionThe best schedule, in which maize production wasprofitable every year, was searched by calculating theSEV for many combinations of planting density (N)and prior-thinning between-tree competition level

(Thin-CItree). The thinning percentage (Thin-%) wasalways 13% and the rotation length (R) was 20 years.A planting density of 1000 trees per ha and a be-tween-tree competition level of 0.21 gave the highestsoil expectation value (Table 3).

The SEV for this management schedule is 11497USD ha−1 and the net annual income is 854 USDha−1. The mean annual maize grain yield is 1620kg ha−1 and the mean annual wood production is 24m3 ha−1. This management schedule recommendsthat the stand should be thinned five times at years13, 14, 16, 17 and 19 (Figure 4). A comparison of thismanagement schedule to the non-constrained opti-mum reveals that the requirement of continuouslyprofitable maize production decreases the mean an-nual wood production by 57%, and both the incomeand SEV by 45%. However, the best schedule withthe requirement of profitable maize production is onlyan approximate estimate of the optimum since only avery small fraction of all the possible combinationsof DVs (N, Thin-CItree, Thin-% and R) was tested.

Uneven-aged management

Uneven-aged management without the requirementof profitable maize productionThe best solution for uneven-aged management witha 3-year cutting cycle suggests that the optimal prior-thinning stocking (N) is 2874 trees ha−1; the Weibullparameters are b = 15.24 and c = 2.16; harvestednumbers of trees are H1 = 30 trees ha−1 (for dbh class5–9.99 cm), H2 = 44 trees ha−1 (10–19.99 cm) andH3 = 489 trees ha−1 (20 cm or more); and 563 treesha−1 (Nnew) to be planted after the selection thinning(Table 4). The five best solutions are fairly similar,except the fourth one, which suggests higher stock-ing, higher numbers of both harvested and planted

Table 1. Five best solutions for the even-aged management of maize – G. robusta field in Kenya.

Decision variablesa SEV (USD ha−1) Income (USD ha−1 a−1) Wood (m3 ha−1 a−1) Maize (yrs)b

N (ha−1) Thin-CItree Thin-% R (yrs)

6475 0.29 14.03 20 20776 1547 56 1

6576 0.33 12.79 19 20645 1557 57 1

6363 0.35 12.93 20 20415 1576 57 1

6487 0.23 12.62 21 20286 1543 55 1

5229 0.34 12.72 20 20245 1528 53 2

aDecision variables are tree planting density (N), between-tree competition level that induces a thinning treatment (Thin-CItree), thinningpercentage (Thin-%) and rotation length (R).b The number of years per rotation when maize production is profitable.

Figure 4. Development of stand volume in the non-constrainedoptimum for even-aged management system and in the optimumin which maize production must be profitable every year.

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trees, and smaller average tree size than the other so-lutions do (Figure 5, Table 4).

The mean annual net income of the optimal solu-tion is about 1650 USD ha−1 and the mean annualwood production is 52 m3 ha−1 (Table 4). Maize pro-duction is never profitable in the management systemwhich maximises the mean annual net income. Fig-ure 6 and Table 5 reveal that the final stand of the bestsolution is very similar to the initial stand, whichmeans that the income and wood production levels ofthe optimal solution are nearly sustainable. However,because the final stocking is slightly less than the ini-tial one (Table 5), a small downward correctionshould be made in the income and wood productionestimates. Differences between the initial and final

Table 2. Sensitivity of the soil expectation value (SEV, USD ha−1) of an even-aged G. robusta stand to changes in decision variables (DVs).

DVa −30% −20% − 10% Optimum +10% +20% +30%

N (ha−1) 19 554 19 233 19 951 20 776 19 360 19 841 20 143

Thin-CItree 17 759 19 293 20 184 20 776 19 924 19 403 19 254

Thin-% 19 504 20 570 20 486 20 776 19 318 19 210 18 227

R (yrs) 18 483 19 774 20 250 20 776 19 747 19 362 18 935

a Decision variables are tree planting density (N), between-tree competition level that induces a thinning treatment (Thin-CItree), thinningpercentage (Thin-%) and rotation length (R). The optimal values of all the four DVs were individually increased or reduced by 10, 20 and30% after which the SEV was calculated.

Table 3. Maximum competition level inducing a thinning treatment (Thin-CItree) for varying planting densities (N), a 20-year rotation (R)and a 13-% thinning percentage (Thin-%), when profitable maize production is required in every year.

N (ha−1) Thin-CItree SEV (USD ha−1) Income (USD ha−1a−1) Wood (m3 ha−1a−1)

500 0.5c 8220 618 16

1000 0.21 11497 854 24

1500 0.13 10416 733 22

2000 0.10 10370 737 23

2500 0.05 9665 694 22

SEV = soil expectation value. Income = mean annual net income. Wood = mean annual wood harvest.c The limit for natural mortality.

Table 4. Five best solutions for the uneven-aged management without the constraint that maize production must be probitable every year. Nis the number of trees per ha, b and c are Weibull parameters (Equation 2), H1, H2 and H3 are the number of trees per ha harvested fromdiameter class 5–9.99cm, 10–19.99 cm and 20 cm or more, respectively, and Nnew is the number of new trees per ha planted after thinning.Maize production was never profitable in any of the solutions. OF = objective function value, Income = mean annual net income, and Wood= mean annual wood harvest.

Decision variables OF Income (USD ha−1 a−1) Wood (m3 ha−1 a−1)

N (ha−1) b c H1 (ha−1) H2 (ha−1) H3 (ha−1) Nnew (ha−1)

2874 15.24 2.16 30 44 489 563 1319 1652 52

2399 16.94 2.15 0 0 416 419 1294 1579 49

2437 16.65 2.05 0 36 400 428 1240 1512 48

3706 14.33 2.34 0 35 534 867 1237 1483 48

2406 16.25 1.97 0 27 367 442 1194 1449 46

Figure 5. Diameter distributions of the initial stand in the five bestnon-constrained solutions for uneven-aged management system.

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stands also mean that some more trees should beplanted and slightly fewer trees harvested than theoptimal solution suggests.

Uneven-aged management with a requirement ofprofitable maize productionWhen a penalty of 1000 USD ha−1 was subtractedfrom the objective function in every year when maizeproduction was not profitable, the optimal combina-tion of decision variables was clearly different fromthe non-constrained optimum (Table 6). The optimalprior-thinning stocking (N) was 1336 trees ha−1; theWeibull parameters were b = 18.39 and c = 1.85; har-vested numbers of trees were H1 = 0, H2 = 0 and H3= 270 trees ha−1; and 170 trees ha−1 (Nnew) were tobe planted after the selection thinning (Table 6). Themean annual wood harvest was 38 m3 ha−1. The num-ber of trees planted after selection thinning (170 trees

ha−1) was 100 trees ha−1 less than the harvested num-ber of trees. Forcing profitable maize production inthe solution decreased the wood production by 27%and the mean annual net income by 24% (Tables 4and 6).

The pre-thinning diameter distributions of the bestsolutions, as described by the Weibull function, areclose to each other in the five best solutions, exceptthe fourth best solution (Figure 7). The deviance ofthe fourth best solution from the other good solutionscan also be seen from Tables 6 and 7.

Comparison of the initial and final stands of thebest solutions reveals that the solutions do not com-pletely fulfil the criterion that the final stand shouldbe similar to the initial stand (Table 7). Because thefinal volume is usually somewhat lower than the ini-tial volume, and the number of harvested trees islarger than the number of planted new trees, the har-vests should be slightly less than the solutions sug-gest, and some more than 170 new trees should beplanted after the cutting. After these adjustments, thesustainable net annual income would be around 1100USD ha−1, and the sustainable annual wood produc-tion about 33 m3 ha−1.

In the best solution, the final number of trees inthose diameter classes which were used in the pen-alty function and in the calculation of income fromtimber sales is nearly equal to the initial number inthe three largest classes (Figure 8). The main differ-ence between the initial and final distributions is thatthere are no trees smaller than 5 cm in the final stand.This means that all trees which were planted after thethinning had grown thicker than 5 cm in three years.

Discussion

This study was the first attempt to optimise the man-agement of Grevillea robusta – maize agroforestrysystem. The results should be considered as prelimi-nary for many reasons. The primary reason is that themodels used by the simulation program are based ona rather small material, measured from temporarysample plots (Muchiri et al. 2001a (in press, 2001b)).The most obvious shortcoming of the models is thatthey may not be reliable for dense stands of G. ro-busta because the modelling data set represented typ-ical stockings of current agroforestry fields. However,because both the tree diameter model and the maizebiomass model were distance-dependent, and themodelling data set of Muchiri et al. (2001a) (in press,

Table 5. Initial (Ini) and final (Fin) values of some stand charac-teristics in the five best non-constrained solutions for the uneven-aged management of maize – G. robusta field in Kenya.

N (ha−1) G (m2 ha−1) V (m3 ha−1) Dg (cm) Dn (cm)

Ini Fin Ini Fin Ini Fin Ini Fin Ini Fin

2874 2712 52 52 295 289 19 18 14 15

2399 2272 51 51 308 304 21 20 15 16

2437 2304 51 51 307 309 21 21 15 16

3706 3492 58 61 312 330 18 17 13 14

2406 2308 49 50 297 300 21 20 15 15

N = the number of stems. G = stand basal area. V = stem volume.Dg = mean diameter weighted by tree basal area. Dn = mean di-ameter.

Figure 6. Number of trees per hectare in the non-constrained op-timum in the five diameter classes used in the penalty function.Initial = initial prior-thinning stand; Final = stand at the end of a3-year cutting cycle.

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2001b) included some very dense groups of trees, thesimulations of this study cannot be said to be extrap-olations. Another clear shortcoming of the availablemodel set is the lack of empirical tree mortalitymodel. In the absence of this model, we had to relyon an ad hoc survivor function, which was based ona guess of the highest possible competition level foran average tree.

The search for the best even-aged managementschedule with the requirement that maize productionmust be profitable in every year was not based on op-timisation but on simple trial and error. A fixed har-vest percentage (13%) and a constant rotation length(20 years) were used in these calculations. Becauseof this, the best constrained schedule reported foreven-aged management may be relatively far from thetrue optimum. Another shortcoming of the calcula-tions for even-aged management system is that theprior-thinning between-tree competition level and theharvest percentage were the same through the wholerotation. The reason for constant thinning parameters

was that this greatly simplified the optimisation prob-lem. Better objective function values may be achievedwhen the thinning parameters are allowed to changefrom thinning to thinning. Especially with the con-straint that maize production must be profitable everyyear, it would most probably be worthwhile to in-crease the prior-thinning competition level towardsthe end of the rotation. This is because tree growthand maize yield depend on competition by trees indifferent ways: maize yield is sensitive to the numberof competitors and tree growth to the size and prox-imity of competitors.

Because of the preliminary nature of the study, thesensitivity of the solutions to the economic parame-ters (production costs and selling prices) and dis-counting rate was not tested. However, most of thesechanges would be quite obvious. It can for instancebe concluded that increased tree planting costs de-crease optimal planting density but even a substan-tially lower planting density would decrease woodproduction only slightly. Higher discounting rateswould obviously result in lower stand densities: plant-ing densities would be lower, thinnings heavier, androtations shorter.

Comparisons of the best solutions and the insensi-tivity of the objective variable to changes in decisionvariables indicate that the objective functions are flatfunctions of DVs near the optimum. Small deviationsfrom the optimal values of management parametershave only a negligible effect on the objective varia-ble. This means that it is enough to specify the opti-mal management regimes only approximately.

The general conclusion that can be drawn from theresults of this study is that economically optimal man-agement means concentrating on wood production. Inthe non-constrained optimum for even-aged forestrymaize production was profitable only during the first

Table 6. Five best solutions for the uneven-aged management with the constraint that maize production must be profitable every year. N isthe number of trees per ha, b and c are Weibull parameters (Eqn. 2), H1, H2 and H3 are the number of trees per ha harvested from diameterclass 5–9.99cm, 10–19.99cm and 20cm or more, respectively, and Nnew is the number of new trees per ha planted after selection thinning.OF = objective function value, Income = mean annual net income, and Wood = mean annual wood harvest.

Decision variables OF Income (USD ha−1 a−1) Wood (m3 ha−1 a−1)

N (ha−1) b c H1 (ha−1) H2 (ha−1) H3 (ha−1) Nnew (ha−1)

1336 18.39 1.85 0 0 270 170 1016 1256 38

1352 18.39 1.98 0 0 292 186 997 1326 40

1256 19.40 1.92 0 0 271 170 990 1231 37

1176 22.29 1.77 0 0 206 130 964 1066 32

1228 19.48 2.12 0 0 272 170 956 1232 37

Figure 7. Diameter distributions of the initial stand in the five bestsolutions, which were constrained so that profitable maize produc-tion was required in every year.

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year after tree planting. In uneven-aged forestrymaize production was never profitable. Because thedifferences between non-constrained and constrainedoptima in objective function value were rather large,it can be concluded that a substantial increase inmaize grain price or decrease in timber price isneeded before a joint production of wood and maizein the same field becomes the most profitable choice.However, it is equally clear that if all farmers beginto produce only wood, the price ratio between maizeand timber will change immediately and the joint pro-duction or even pure maize production without anytrees becomes again the best alternative. The resultsof this study should be understood so that, with thecurrent prices, it would be profitable for an individualfarmer to concentrate on wood production if the otherfarmers do not follow the example.

An explanation for the apparently non-optimalpractices with the emphasis on maize production isthat maize production on the farmer’s own land is forfood security rather than for profit. Tree growing may

be more often a purely economic activity, but it isconsidered less important than producing staple foodfor the family. The farmers may also realise that in-creased wood production on a larger scale would soondecrease wood prices.

The results of this study cannot be used for com-paring the profitability and mean annual income ofeven-aged and uneven-aged management systems.This is because the optimisations for uneven-agedsystem concerned the period when the stand alreadyhad an optimal steady-state structure, whereas the cal-culations for even-aged management began with theestablishment of a tree stand in an open area. The ob-jective function values of the uneven-aged manage-ment system did not include the establishment cost ofthe initial stand but this cost was included in all cal-culations for the even-aged management system.

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