1. LimingMaterial Selection by ComputerSpreadsheet
Zhi-ChengLi, I P. G. Widjaja-Adhi, T. S Dierolf, and R. S. Yost*
ABSTRACT
Limeselection considerationsare complexandmaybe sim-
plified withthe aid of a computerspreadsheetwithoptimiza-
tion capability. A mathematicalmodelwasformulatedusing
linear programmingtechniquesthat includedlime quantity,
limingdepth,limecost, limequality(calciumcarbonateequiv-
alenceandfineness), amountreactedafter 3 too, Cacontent,
Mgcontent,distanceof transport,andtransportationcost per
unit distancefor eachof five limingmaterials.Programresults
weretested withsituations anddatafrompreviousexperience
in Sitiung,Indonesia.Thepredictionandassociatedsensitivity
analysiswasuseful in planninglimingoperationsandshould
providea usefultool to explorelimemanagementoptions. The
spreadsheetmodelshouldaid learningaboutthe variouscom-
ponentsandcomplicatingfactors importantin selectingliming
materialsandtechniquesto mergethe considerationsinto a
single lime recommendation.
SOILACIDITYCONTINUESto limit agricultural productivity
in the tropics becauseof inherent acidity and intensive
cropping. Although modemapproaches to acidity manage-
mentinclude acid tolerant germplasmand, in somecases,
the use of organic materials to reduceacidity, the needfor
limestoneneutralization of acidity continues.
Anestimated 33%of soils of the tropics are Oxisolsand
Ultisols, manyof whichneedlimestoneapplications to alle-
viate yield restricting effects of acidity (Sanchez, 1976).
Limerecommendationsfor highly weathered tropical soils
are commonlybased on reducing soil aluminumsaturation,
(extractable Al + H)/(effective cation exchangecapacity)
100, belowa critical level for a given crop. TheAcidity
Decision Support System (ADSS)(TropSoils, 1991) is
expert systemthat determineslime requirementsfor various
crops grownon highly weatheredtropical soils basedon Al
saturation of the soil and Al tolerance of the crop. However,
the process of makinglime recommendationsis complicat-
ed by additional factors such as maintainingadequatesoil
Caand Mglevels within a range of Ca/Mgratios (Souzaand
Ritchey,1988;Cochrane,1989),cost andavailability of lim-
ing and fertilizer materials, andcost of transportation. The
averageindividual wouldhavedifficulty accountingfor all
these factors while attempting to minimizecost. Weshow,
however, that spreadsheet and linear programmingtools
available in manycommerciallyavailable spreadsheets such
as Quattro Pro, Excel, and Lotus 1-2-3, can simplify this
problem.
Usually Ca/Mgratios have little importance whensoil
acidity is neutralized and sufficient Caand Mglevels are
Z.-C.Li, Univ.of Hawaii,1910EastWestRoad,Honolulu,HI96822;I
P.G.Widjaja-Adhi,Centrefor Soil andAgroclimateResearch,Bogor,
Indonesia;T.SDierolf,JalanKehakimanno.283,BukitTinggi,Sumatra,
Indonesia;andR.S.Yost,Univ.of Hawaii,1910EastWestRoad,Honolulu,
HI96822.HawaiiInst. for TropicalAgile.andHumanResourcesTechnical
PaperNo.4117.Received26 July 1994.’Correspondingauthor
( rsyost@hawaii.edu).
Publishedin J. Nat.Resour.LifeSci.Educ.25:26-30(1996).
ensured (Lopes, 1983; Souzaand Ritchey, 1988; Cochrane,
1989). Nevertheless, someacid soils of the tropics contain
such small amounts of Ca and Mgthat imbalances can
occur. In the Brazilian Cerradothe least productivevegeta-
tion group, cerrados, was characterized by Ca/Mgratios
<1.0, whereasforest vegetation wascharacterized by ratios
of >_3.0(Cochrane,1989). Onan acid Oxisol, neutralization
of acidity with MgCO3 resulted in soybeangrain yields of
less than 100 kg ha-t while neutralization with CaCO3 pro-
duced average yields of 2100kg ha-t (Souza and Ritchey,
1988). Theseauthors suggested that lowyields whereacid-
ity wasneutralized with MgCO3 wasdue to excessive soil
Mgin relation to soil Cathat resulted in a Ca/Mgratio <1.0.
Onthe other hand, relying solely on calcitic limestone to
alleviate soil acidity maynot provide the necessary mini-
mumlevel of Mg.
Transportationandapplication costs are interrelated with
limequality. Moreof a less effective material mustbe trans-
ported and applied to neutralize the soil acidity compared
witha highly effective material. Similarly, the Mgrequire-
mentsmaydictate use of a dolomiticlime while neutralizing
the desired amountof acidity. Becausedolomitic limestone
maybe moreexpensiveand less reactive than calcitic lime-
stone, it maybe moreeconomicalto apply only the amount
of dolomitic lime necessary to supply Mgand use calcitic
lime to neutralize the remainingacidity. Other cases may
require the applicationof caicitic limewithfertilizer sources
of Mgthat have no liming value such as MgSO4.
The problem becomescompoundedwhenthere are sev-
eral materials available, alone or in combination,that can
both alleviate soil acidity as well as providesufficient Ca
and Mglevels. In somecases, available materials cannot
both neutralize acidity and provide sufficient Ca and Mg
levels, even within the wide range of acceptable Ca/Mg
ratios. Thesematerialsusuallyalso differ in price, availabil-
ity, andquality.
Thus,selecting the best liming materials and determining
their correct proportions,whileconsideringa variety of con-
strain~, can be a perplexing task. A simplex linear pro-
gramhf~agalgorithm (Dantzig, 1951) can performthis type
of optimization as long as the problemcan be specified
numerically.
In this article, wedescribe the formulationof a mathe-
matical modelthat can solve the problem. Next, weshow
howthe model can be used with a commonlyavailable
spreadsheet program.Finally, wewill showapplications of
the modelto a decision-makingscenario in Indonesia.
MATERIALS AND METHODS
Model Formulation
Consider a scenario in which a farmer’s field has been
sampled, the soil tested, and a lime requirement must be
given. Alime requirementcan be obtained froma variety of
Abbreviations:ADSS,AcidityDecisionSuppot~System;ECEC,effective
cationexchangecapacity;BD,bulkdensity.
26 ¯ d. Nat.Resour.Life Sci. Educ.,Vol. 25, no.1, 1996

2. lime recommendation software including the ADSS
(TropSoils, 1991). This softwarewill estimate lime require-
mentsfor soils with AIas the majorlimiting factor, provid-
ed the user has the following minimumdataset: AI+Hcon-
tent of the soil, effective cation exchangecapacity (ECEC)
(sumof KCl-extractable Ca, Mg,K, Na, and AI+ H), soil
bulk density (BD),desired crop, lime quality (calciumcar-
bonateequivalencyand particle-size distribution), and man-
agementoptions such as intended depth of lime incorpora-
tion andquantity of organicmaterialsapplied.
This recommendation,however, is somewhatincomplete
becausethere are various sources of limestone, whichvary
in cost, neutralization quality, and content of Caand Mg.A
fully specified recommendationcan include the following
requirements:
1. Sufficient limeto neutralize excessiveacidity on a per-
ha basis
2. Considerationof the total quantities of limingmateri-
als available
3. Consideration of the Ca and Mgrequirements to pro-
vide critical amountsof Caand Mgin the soil, and a
desireable soil Ca/Mgratio after the liming material
has reactedwiththe soil
4. Determinationof the lowest cost combinationof lim-
ing materials and other Caand Mgcontaining materi-
als in relation to economicreturn
5. Consideration of Ca and Mgmaterials that have no
liming valueas longas they contribute to satisfactory
Ca, critical amountsof Mg,and Ca/Mgratios that
remainwithin the desired range
Computation
Thelinear programmingframeworkfor this problempro-
ceeds as follows. Theobjective function computesthe total
cost of limingmaterialsfromfive possible materials that we
haveincluded. (The user can expandthis to consider more
liming materials). Theobjective function is minimizedsub-
ject to various constraints, whichare described aboveand
illustrated in Table1. Wenowdescribe somedetails in for-
mulatingthe constraints for the mathematicalmodel.
Theeffect of lime applicationon critical amountsof Caand
Mg,andon desired Ca/Mgratio after liming.
1. Initial Caand Mg:
Cao = initial soil Ca,cmole/kg
Mgo = initial soil Mg,cmol~/kg
2. Theincrease in Ca and Mgfrom reacted lime material
i (i = 1, 2, 3, 4, 5) as measuredin increased 1 MKCI-
extractable Caand Mg,cmolc/kg:
Cai = (ci xi ri)/(A x BDx Depthx 20)
whereci is the amountof Cain the limingmaterial, %;xi is
the amountof limingmaterialfromsourcei, kg; r i is the pro-
portionof the limingmaterialthat will react in 3 mo(default
is 0.667);Ais the area of the parcel, ha; BDis the bulkden-
sity (from0 to 2), g/cm3;Depthis the intendeddepthof lime
incorporation, cm; and the value 20 derives from 20 kg
Ca/ha-cm, which is 1 cmolc Ca/kg if BD= 1.0 g/cm3 and
depth is 1 cm(note: this is modified based on actual BD
value above).
Mgi = (mi Xi ri)/(A x BDx Depthx 12)
wheremi is the %Mgin the liming material and the other
terms are defined abovefor Ca, and the value 12 derives
from 12 kg Mg/ha-cm,which is 1 cmolc Mg/kg.
3. Theaboveexpressionsare then used to derive the con-
straints for the minimumcost objective function.
Cao + Z~= t Cai -> Kca [1]
Mgo + g5. ~=1 Mgi> KMg [2]
1 < [Cao +y.5/= ~ Cai]/ [Mg° + E~=~ Mgi]< F
[3]
wherethe terms are as described aboveexcept that Kcaand
K.. are critical levels of Caand Mg,respectively, andF isMg
the maximumallowable ratio of Ca/Mgin cmolc/kg, usual-
ly set at 8 to 10.
Toensurean adequateamountof lime to neutralize acidity,
the followingconstraintis establishect
g~= 1 xi CCEifii = QA [4]
wherexi is defined as above;CCEi is the fractional expres-
sion of calciumcarbonateequivalenceof the liming materi-
al; andfi is the finenessfactor of the limingmaterial,calcu-
lated as the sumof twofractions: (i) the proportionof lime
that is between30 and 50 mesh,considered10%effective in
acid soils of the tropics (Alcarde, 1983;Barber,1984); and
(ii) the proportion of lime finer than 50 mesh,considered
90%effective (Alcarde, 1983); Qis the lime requirement,
tonnes/ha;andAis the parcel area consideredfor liming,ha.
Data Requirements
1. Soil data including exchangeableCa, Mg,and soil BD
2. Depthof intended lime incorporation
3. Analysis of lime materials: CaCO3 equivalence and
particle-size distribution
4. Cost of limingmaterial at the purchasepoint, includ-
ing the distance and transportation cost per unit
weightand per unit distance
5. Amountof Caand Mgmaterials available for applica-
tion
6. Critical levels of exchangeable Ca and Mgand the
maximumallowable of soil Ca/Mgratio
Spreadsheet Formulationof the Model
This modelcan be applied using almost any spreadsheet
software withlinear programmingcapabilities (e.g., Quattro
Pro, Excel, and Lotus 1-2-3). Thefollowing exampleshows
howthe modelwas used in Ouattro Pro 5.0 Windows.The
modelwasalso implementedin Quattro Pro 4.0.
Ourspreadsheetdesign contains three sections (Table 1).
Theinput data section is usedto enter data required to run
the model.Theinternal calculations section includes inter-
J. Nat.Resour.Life Sci. Educ.,Vol. 25, no.1, 1996¯ 27

3. Table 1. Spreadsheetdesign for using the optimizerroutine in QuattroPro5.0 to solve the liming problem.(costs in $US).
R
0
W
S
OI *
02
O3
O4
O5
O6
07
O8
O9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
35
36
37
38
39
41
42
43
44
4.5
46
47
48
49
50
51
Spreadsheetcolumns:
A B C D E
Soil Characteristics
Location ~ Initial soil Ca(cmolcAg) 0.83
Area(ha) 100 Initial soil Mg(cmolc/kg) 0.13
Limerequirement(tonne/ha) 3.5 Minimumdesired Ca/Mgratio /
Soilbulkdensity(g/cm3)
1.00 Maximumdesired Ca/Mgratio 8
Incorporationdepth(cm) 15 Criticalsoil Ca(cmolc/kg) 200
Criticalsoil Mg(cmolc/kg) 0.25
Lime/fertili:cffmaterial
Materialcharacteristics
I. CCE(%):Calciumcarbonateequivalence(CaCO3 = 100%)
2. Ca(%):Cacontentof the material
3. Mg(%):Mgcontentof the material
4. Finenessfactor (calculatedas 4a+4b):(nonlimingmaterial=O)
a. Proportionof limebetween30and50mesh(0-1)
b. Proportionof limefiner than50mesh(0-1)
.5.Reactionfactor (fractionfrom0-1):fractionof applied
materialthat reacts withinthreemonths
Costsandavailability
6. Costof material,notincludingtransportation(price/tonne)
7. Totaltransportationcosts(price/tonneXcslculatedss 7a x7’0)
a. Distance(kin)
b. Transportationcost(price/tonne/km)
8. Othercosts(price/tonne)
9. Totalcost(prico/tonoe)(calculatedas 6+7+8)
10.Availabilityof eachmaterial(tonnes)
z=-l/(Areax Incorporationdepthx Soilbulkdensity
l 2 3 4 5
1.0 1.0 1.0 1.0 0.0
23.81
1.20
25.01
* Internal calculations * *
z : 0.0007
Ca
i
: 0.0061
Mgi: 0.0003
23.81 21.43 26.19 142.86
1.20 10.00 10.00 1.20
25.01 31.43 36.19 144.06
0.0007 0.0007 0.0007 0.0007
0.0017 0.0073 0.0036 0.0
0.0022 0.0 0.0024 0.0053
Finenessfactor x CCE: 0.85 0.85
Totaleffectivelime(tonncs) 350
Totallimerequirementremaining(tonnes) 0
* * * * * * * * * * * * * * * * * * * Solution * * * * * *
Cost
Totalcost 11077
Cost/ha 110.77
Amountof materials requital
1 2
Total amount of each material required (tonnes) 100 50
tonnes/ha 1.00 0.50
Material still available (tonnes) 0 0
Predictedsoil properties
SoilCaafter liming(cmoIJkg) 3.01
SoilMgafter liming(cmolJkg) 0.38
Predictedsoil Ca/Mgratio 8.0
0.99 0.96 0.00
Lime/fertilizermaterial
3 4 5
180 46 0
1.80 0.46 0.00
9820 9954 3
mediate results that the modeluses to determine final
results. Thefinal results are shownin the solution section
and include minimizedtotal cost, optimizedamountof each
material required, and predicted soil Caand Mglevels that
would result from the application of the optimummixof
liming and Ca- and Mg-containingmaterials.
Threecell types are indicated in the spreadsheet. Input
cells (identified byunderlining) are used for entering data.
Outputcells (bold face) contain calculations conducted
the input data. Textcells (normaltext face) are usedto iden-
tify the contents of input andoutputcells.
Thespecific formulasare listed in Table2 together with
a brief explanation of the calculation performedby each.
Theformulasare so simpleand straightforward that it seems
unnecessaryto distribute the binary formof the spreadsheet.
If readers prefer to obtain copies of the spreadsheet, it is
available by sendingone formatted3.5-inch diskette to R. S.
Yostat the University of Hawaii.
This spreadsheetdesign can be duplicated easily for han-
dling morethan five liming materials. Text cells shouldbe
filled with the appropriatetext shownin Table1. Input cells
canbe left emptyinitially andfilled later withthe appropri-
ate data. Outputcells contain formulas, whichare shownin
Table 2. Note that the copy commandshould be used to copy
someof the formulas from ColumnB to ColumnsC through
E
Theoptimizer routine uses linear and nonlinear program-
mingtechniques to determine the optimumsolution. This
routine requires the identification of a solution cell, variable
cells, andconstraints (Table3). Select minimizein the solu-
tion cell windowto find the lowest cost solution. Themodel
optionshouldbe set to nonlinearbecauseof the effect of Eq.
[3].
28¯ J. Nat. Resour.Life ScLEduc.,VoL25, no. 1, 1996

4. Table2. Formulasrequi~lfor outputcells in thespreadsheetdesign.
Spreadsheetcellformulas
Formulatoenterincell: Theformulacalculates:
Onecell formulasto enterin columnBonly
B35:(B44* B34+C44* C34+D44
¯ D34+E44* E34+F44*F34)
B36:@ABS(($B$35)-($B$5* $B$4))
B39:@SUM(B44* B26,C44* C26,D44
¯ D26,E44* E26,F44* F26)
B40:(B39/$B$4)
B49:(F4+B44* B30+C44* C30+D44* D30
+E44* E30+F44* F30)
B50:(F5+B44* B31+C44* C31+D44* D31
+E44* E31+F44* 1731)
B51:(B49/B50)
Totaleffectivelime
Totallimerequirement
remaining
Tntalcost
Cost/ha
SoilCaafterliming
SoilMgafter liming
PredictedCa/Mgratio
Formulasto enterin columnBandcopyto columnsC,D,E,andF
BI4:(BI5+BI6) Finenessfactor
B22:(B23* B24) Totaltransportationcosts
B26:(B21+B22+B25) Totalcost
B29:I/($B$4* $1753* $F$6) z
B30:(5* B29/14)* BI2* BI7 Cai
B31:(B29/2)* B13* BI7 Mg~
B34:(Bll* BI4) Finenessfactor x CCE
!745:(B44/$B$4) tonncs/ha
B46:(B2.7-B44) Materialstill remaining
Table3. Parametersandformatsfor the optimizerroutine.
Parameter Format
Solutioncell
Variablecells
Constrainls
B39..B39
Minimize
B51>=B6
BSI<=B7
B36--0
B44..F44=>O
B46..F46>=O
Options Nonlinear
Newton
RESULTS
Decision-Making Applications
Weapplied this model in Sitiung, Indonesia. The major-
ity of farms in the Sitiung area are located on highly weath-
ered Ultisois and Oxisols where AI toxicity is a major con-
straint to annual food crop production. Thesoil AI saturation
was found to be a reliable indicator of crop response to lim-
ing and critical AI saturation levels have been determined
for several annual food crops (Wadeet al., 1988).
For this application, several farmer groups in a village
determine that they wish to purchase lime for their fields.
Although average farm size is only 1 ha, the total amountof
land to be limed is 100 ha. Wewill assume that extension
agents have sampled many of the farms and used the ADSS
to recommendan average of 3.5 tonnes of lime/ha. Wealso
have to assume that several soil samples were taken to mea-
sure the bulk density and exchangeable Ca and Mg, and we
used the average results of these samples in our application.
The village can either purchase lime already locally
available or from a factory located 200 kmfrom the village.
Thefive materials indicated in Table 1 showthe quality and
cost for the various materials available to the village. The
first two materials are locally available calcite and dolomite
leftover from a government liming program, the third and
Table4. Effectof changinginitial parametersonsolution.
Solution
Amountof eachmaterial
Parameter
changed Changesmade t 2 3 4 5 Cost
Original None(seeresultsinTable!)
Ca/Mgratio Decreasedmaximumfrom8 to 3
Areato belimedDecreasedfrom100to 10ha
TransportationDecreasedfrom200to 100km
distance formaterials3and4
~ tonnes ~ (S/ha)
100 50 182 45 0 110.70
100 50 13 218 0 120.98
33 8 0 0 0 102.98
0 50 258 54 0 97.61
fourth materials are available at a factory located 200 km
away, and the last material is locally available kieserite
(MgSO4)(Table
The solution to this problem is shown in Table 1. Note
that all of the locally available liming materials (no. 1 and 2)
were used. Because of limited availability of the first two
materials, additional materials will have to be purchased
from the factory.
Sensitivity analyses were conducted to determine the
effect of a more restrictive Ca/Mgratio, liming less land,
and the effects of a closer lime source. In the original exam-
pie, Ca/Mgratio was allowed to range between 1 and 8. The
effect of changing the maximumCa/Mg ratio from 8 to 3
results in an increase in cost per hectare, with moreof mate-
rial no. 4 required to satisfy the lower maximumCa/Mg
ratio (Table 4). Note that in the original example the Ca/Mg
ratio was 8 for the minimum cost solution (Table 1).
Decreasing total area limed from 100 to 10 ha resulted in a
decrease in cost per hectare because all of the materials
could be purchased locally (Table 4). For the final example,
transportation distance is decreased from 200 to 100 km,
which may result if a factory is established closer to the
Sitiung area. This change resulted in a preference for mate-
rials no. 3 and 4 over material no. 1 because it was more
economical to obtain these materials from the new factory
(Table 4). This option also resulted in the lowest cost per
hectare for any of the four options, illustrating the beneficial
impact of the newlime plant. The high cost of kieserite (a
Mgsulfate product) precluded its selection for any of the
options.
DISCUSSION
In developing this model of limestone application we
used estimates of lime effectiveness (calcium carbonate
equivalence and reactivity as predicted by the particle-size
distribution of the liming material) and standard agronomic
techniques. In formulating the mathematical model for opti-
mal liming two important results became apparent: (i) stan-
dard agronomic calculations of lime rate, Ca and Mgcontent
were very easy to express in linear programmingform; and
(ii) rate of lime reaction with soil is critical to estimating the
resultant Ca and Mgcontents of the limed soil; yet this reac-
tion rate and factors that affect it are surprisingly not fre-
quently measured in liming research.
Quantitative estimates of lime reaction rate would help
improve lime management. Factors important to estimating
reactivity include: soil pH, nature of acidity, water content
and drainage, soil solution conductivity, lime quality and
size distribution, cropping intensity, N source, and rate.
J. NaLResour. Life ScL Educ., Vol. 25, no. 1, 1996¯ 29

5. This application illustrates how commonly available
software can be a powerful decision-making tool in agricul-
tural lime source selection.
ACKNOWLEDGMENTS
The authors would like to recognize the stimulatingdis-
cussions with Mr.Edson Lobato and Djalma M.G. de Souza
of the Brazilian Cerrado Center, Brasilia, Brazil, that moti-
vated this work.