Forest Science,Vol. 34, No. 3, pp. 647-661.

Copyright1988by the Societyof AmericanForesters

A Markov Chain Model for Evaluating SeasonalForest Fire Fighter Requirements DENNIS DAVID

BOYCHUK L. MARTELL

ABSTRACT. A model was developedto help resolve the decisionof how many fire fightersa large forestfire managementagencyshouldhire for a fire seasonto minimize

expectedcost plusfire loss. It addresses the use of fire fightersfor both initial and extendedattack,the temporaryhiringof extrafire fightersto satisfypeakdemands,and the movementof fire fightersbetweenregionswithin the agency'sjurisdiction.It also recognizes the increase in subsequentfire fighter demand caused by fire fighter shortages.The fire management systemis modeledas a Markov chainwherethe system state is relatedto the fuel moistureconditionsthroughoutthe protectedarea, and the total number of extendedattack fires burning.Testsusingrepresentativehypothetical data demonstratedthat it is a tractableapproachfor roughlyevaluatingfire fighterrequirements.The model was used to evaluate the benefits of the centralized control of fire fighters,andindicatedthat it couldreduceexpectedcostpluslossas muchas30% in the hypotheticalexample. FOR. ScI. 34(3):647-661. ADDITIONALKEY WORDS. Forestfire management,decisionmaking,economicforest protection.

EACH YEAR,FORESTFIREMANAGERS in the province of Ontario must decide how many "regular" fire fightersto hire for the fire season.This decisionis made before the seasonstarts and is normally not revised as the season progresses.A numberof factorscomplicatethe decision: 1. There is a large year-to-year and week-to-week variation of fire activity that

cannotcurrentlybe forecastmore than a few days, muchlessa few monthsin advance.In recentyears,the numberof firesin Ontariohasrangedfrom approximately 1,000to 2,000 per year, and the burned area has rangedfrom about 5,000 to 500,000 ha per year (Ontario Ministry of Natural Resources1986). 2. Since "temporary" fire fighterscan be hired as requiredduringthe fire season,

it is not necessaryto hire enoughregularfire fightersto satisfypeak demands. While less expensive, temporary fire fighters are not used to satisfy all fire fighter requirements,as they are generallylesseffective, and are usuallyavailable only after some delay.

3. In a largejurisdiction,fire activity canvary considerably from placeto placeon a given day. The ability to temporarilyrelocatefire fightersfrom their home basesto regionsthat are temporarilyexperiencinghigh fire activity reducesthe total numberof fire fightersneeded,but incurstransportationcosts. 4, Interagency agreementsand internationaltreaties make it possibleto borrow trained regularfire fightersfrom other fire managementagenciesin Canadaand the United Statesfor shortperiodsof time. 5.

Fire fighters are used for both initial attack on fire arrivals (i.e., new fires that

The authorsare graduatestudentand associateprofessor,respectively,Facultyof Forestry, University of Toronto, Toronto, CanadaM5S 1A1. This paper is based on D. Boychuk's M.Sc.E thesisresearchconductedat the Faculty of Forestry, University of Toronto. We thank the staff of the Ontario Ministry of Natural Resourcesfor informationconcerningtheir fire

operations.We alsothank M. J. M. Posnerfor a critiqueof the model, and two anonymous refereesand an editorialadvisoryboardmemberfor their helpfulcommentson an earlierversionof the paper.This researchwassupportedby the Natural Sciencesand Engineering ResearchCouncil of Canada. Manuscript received July 7, 1987.

SEPTEMBER 1988/647

are reported), and extendedattack on ongoingfires. Furthermore, fire fighter shortagesincreasethe likelihoodthat fire arrivals will escapeinitial attack and becomelarge. Large fires often create enormousshort-termdemandsfor fire fighters and other suppressionresourcesthat reduce the fire management agency'ssmalland large fire suppressioncapabilities. In additionto the numberof fire fighters,decisionmakersmustchoosethe presuppression hiring level of other resources,betweenwhich there are complex interactions.For example,transporthelicoptersandair tankerscanenhancefire fighter productivity.Changingrelative costsof the resourcesimplieschanging relative proportionsin the optimal mix.

We developeda Markov chain model that explicitly incorporatesall of these factors, with the exceptionof transportationcosts,fire fighter borrowing, and mixes of suppression resources,which we addressin the discussion. Given these exclusions, the major trade-off consideredwhen making the seasonalhiring decisionis the cost of hiring regular fire fighters versusthe cost of large fires and hiring temporaryfire fighters. This is the familiarproblemof choosinga presuppression expenditurelevel to minimize the total of presuppression cost, expectedsuppression cost, and expected net loss (Althaus and Mills 1982). It is also a complicatedversion of the classicoperationalresearch"newsboyproblem" (e.g., see Wagner 1975), which involves choosingthe "stocking level" to meet an uncertaindemand. Our model could be used to roughly evaluate the economicseasonalfire fighter hiring levels for large fire managementagencies,and assessthe economic benefits of centralized control of fire fighters. This paperis organizedasfollows.Relatedpreviousstudiesare described in the following section.An overview of the model is then given, followed by a descriptionof the componentsof the systemstate:fire load and extended attack fire. The "structural model" used to calculate the system statetransitionprobabilitiesis described,first in outline,thenin detailusing the representativehypotheticaldata we usedto test the model. Computational resultsare presented,followedby a discussionof model extensions and limitations.

PREVIOUS

RELATED

STUDIES

Martell (1982) reviewed a numberof studiesthat dealt with seasonalhiring of fire fighters, air tankers, and combinationsof fire-fightingresources. None were specificallydesignedto addressthe seasonalfire fighter hiring decisionfor a jurisdiction as large as Ontario. Gamache (1969) developeda Monte Carlo simulationmodel of a single attack baseto estimateexpectedcostplus lossas a functionof the number of fire fightershired. Fire occurrence,fire characteristics,and suppression start times were randomlygeneratedfrom historicaldistributions.Temporary hiringwas modeled,but interregionalfire-fightertransfersand escaped fires were not. Simard(1973)usedthe systemsdynamicsapproach,a deterministic simulationtechnique,to model air tanker transfersbetweenthree organizationallevels. The number of uncontrolledfires was exogenously specified,and causedair tanker transfers.The total cost of air tankers, transfers,foreign basing,uncontrolledfires, and administrationwas calculated for each simulationrun. Doan (1974) used a stochasticdecisiontree to find the optimaldispatchesand seasonalhiringlevels of initial attack resourcesthat minimizedthe expectedcostplus lossfor a singleattack base. Discrete probabilitydistributionsof fire occurrence,fire characteristics, and suppression force performancewere used.A fixed costand delay were 648/FOREST

SCIENCE

incurredfor usingresourcesfrom outsidethe system.Escapedfires burned only until the end of the day. Greulich (1976) modeled air tanker hiring, home basing, and daily transfer decisionsas a Markov decisionprocess. Daily transitions of a fire danger rating index were modeled as a Markov chain. He maximizedair tanker output subjectto havingexpectedexpenditures satisfya specifiedseasonalbudgetconstraint.Escapedfires and temporaryhiringwere not considered.Quintilioand Anderson(1976)developed a deterministicsimulationmodel of initial attack operationson historical

fires, comparingthe cost and successrates of differenttypesof homogeneoussuppression forces. FOCUS (Brattenet al. 1981)is a detaileddeterministicsimulationmodel that fightshistoricalfires and evaluatesthe cost plus lossof alternativeamountsof mixed fire fightingresourcesand system configurations.Large, escapedfires are evaluated manually, and fought with resourcesfrom outsidethe system. FEES (Mills and Bratten 1982)is a stochastic,nonsite-specificenhancementof FOCUS which evaluates the same types of alternatives, and deals with escapedfires and outside resourcesin a similar way. Except for a small Monte Carlo component,it deals with uncertaintyby evaluatingall combinationsof events defined by discreteprobabilitydistributions.For example,somewhatlike Doan (1974), FEES fights a full set of representativehypotheticalfires with character-

isticsthat are takenfrom discreteprobabilitydistributions. Eachfire is handled independently,so the buildupof workloadcausedby multipleconcurrent fires and consecutiveseveredays is not considered.It explicitly evaluates risk and the impact of fires on multiple forest resourcesover time. Aneja and Parlar (1984)usedstochasticoptimal control theory to model the seasonalfire fighterhiringand dispatchingdecisionfor a very smallsystem with a maximum of two simultaneous fires. Fire occurrence was assumed to

be Poisson,and control time was exponentiallydistributed.Escapedfires and temporary hiring were not considered.Martell et al. (1984) extendedthe work of Quintilio and Anderson (1976) to evaluate initial attack air tanker requirementsfor Ontario.Their deterministicsimulationmodelof suppressionon a sampleof historicalfires addressedcostsand losses,but did not considertemporaryhiring or the use of fire fightingresourceson escaped fires.

A common characteristicof these models is that they do not consider the impact the fires have on the demandfor fire fightingresourcespast the first day of fire suppressionactivity. (Much of the total fire-fighterdemand,and mostof the total suppression and damagecostsare associatedwith escaped fires that burn for two or more days.) Most of the modelsdescribedabove do not adequatelyaddresstemporaryhiring,or the interregionalmobilityof resourceswithin a largejurisdiction.

OVERVIEW

OF MODEL

Figure 1 illustratesthe componentsand generalframework of the model. Briefly, the systemis modeledas a Markov chain, where the systemstate has two components.The first, fire load status, is related to fuel moisture conditionsin eachregionof the province.The second,extendedattack fire status,is the total numberof extendedattack fires burningin the province. The day-to-daytransitionsof the fire load statusdependonly on nature, specificallyweather. The day-to-daytransitionsof the numberof extended attack fires depend on the fire load statusand the number of regular fire fightershired for the season.A structuralmodelof fire fighter deployment and fire suppression activitiesis used to calculatethe transitionprobabiliSEPTEMBER 1988/649

INCOMING SYSTEM

STATE

EXTENDED FIRE LOAD AI-I'AC K FIREll

STATUS TODAIJ STATUS_ TODA•I FIRE LOAD STATUS

MARKOV CHAIN

OUTGOING SYSTEM

STATE

FIRE LOAD STATUS TOMORROW

"STRUCTURAL J

MODEL" OF J DEPLOYMENT ,t,.J SUPPRESSION J

EXTENDED

A'FI'ACK FIRE STATUS TOMORROW

FIGURE1. Overviewof the components of the systemstate,andthe flameworkof the system state transitions.

ties. The Markov chain is discretetime, finite state, first order, and ergodic (e.g., see Howard 1971), from which we calculatethe probability that the system will be in each state. We combine that with the cost and loss associated with being in each state to calculatethe expectedcost plus lossof the specifiedregular fire-fighter hiring level. FIRE

LOAD

The moistureconditionof a forest fuel complexdependslargely on the cumulative effect of weather. The moisture condition, in turn, largely determines(1) the numberof fire arrivals, and (2) the fire behaviorand difficulty of controllingnew and ongoingfires. The fire load componentof the system state representsthe variation over time and spaceof fuel moisture conditions and the associated work load throughout the fire management agency'sjurisdiction. To model the variation over space,the jurisdiction is divided into regions. We used five regionswhich would averageapproximately 180,000km2 in size in Ontario. For a measureof work load in a region,we usedan index from Martell et al. (1984).• Thefire load indexis an estimateof the total flame area (normalized to 1600h)of all fire arrivals in a region in a specific24-hour period. Flame area can be calculatedfrom historicaldata usingfire behavior equations and fire report statistics.For each fire, the perimeter at report time is estimated by assuminga wind influenced shape. The flame length is estimated usingequationsfor fire intensity,fuel consumption,and spreadrate, where the spread rate is adjustedto what it would be at 1600h. The flame area is the flame length times the perimeter and is summedover all fires in the regionto obtain the fire load index for that regionfor the day. More fires, larger fires, or more intense fires lead to a higher fire load • The fire load index is an extensionof the unpublishedfire detectionurgencyconceptdeveloped by P. H..Kourtz of the PetawawaNational ForestryInstitute, CanadianForestryService.

650/FOREST

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index. Hence, the relationship between fire load index, fire arrivals, fire behavior, fuel moisture, difficulty of control, and fire-fighter workload. The fire load indices from all regionsfor all days are ranked, and arbitrarily divided into discretefire load classes.We usedfour: low, moderate, high, and extreme. While representativehypotheticaldata were used here, historical data could be analyzed to determine the distribution of the number of fire arrivals, and fire control successrates for days with different fire load classes.

Regionsare consideredto be "equivalent" if they have (1) approximately the samedistributionof the number of fire arrivals, and difficulty of control, for each fire load class, and (2) approximatelythe same values-at-risk.If significantdifferencesexist, some or all regionsmay need to be treated as "distinct" entities. Our knowledgeof Ontario suggeststhat it can be divided into one distinct southern region, and about four equivalent northern regions. For this study, however, all regions were assumedto be equivalent for simplicity.Given equivalence,thefire load statusfor a day is a specification of the number of regionsin each fire load class. Assumingfive regions and four fire load classes,let the numberof regionsof low, moderate,high, and extreme fire load classbe u, v, w, and x, respectively.The fire load status is

where

u,v,w,x • {0,1,2,3,4,5}, and

u + v + w +x=

5.

For example, fire load status "0221" means no regions are low, two are moderate, two are high, and one is extreme. With five regionsand four fire load classes, there are 56 different fire load statuses. Day-to-day changesof fire load statuscan be modeledas a discretetime, finite state Markov chain. Greulich (1976) found that, in a district in California, the daily transitionsof a fire dangerrating index could be modeled by a first or second order Markov chain, but he used the former due to dimensionality considerations. In unrelated unpublished research, the second author found that, in a district in Ontario, daily transitions of a fire intensity index could be representedby a secondorder Markov chain. We used a first order Markov chain here, but a second order chain could have been used at the expenseof a larger state space. The fire load status is updated at 0900h each day and applies for 24 hours. EXTENDED

ATTACK

FIRES

We wish to model fires that survive and occupyfire fightersfor more than one day. When all of the fire fightersdispatchedto a fire have not returned to an initial attack base in time for dispatchto new fire arrivals by 0900hof the day after initial attack began, that fire is declaredto be an extended attackfire. Extendedattack fires that arisefrom individualregionseach day are added to a running provincial total. The number extinguishedeach day is subtracted. The extended attack fire status is the number of extended attack fires in the provinceat 0900h, and it appliesfor 24 hours.The system

SEPTEMBER 1988/651

state spacecan be kept finite by defininga reasonableupper limit for this parameter.

STRUCTURAL

MODEL

Having definedthe systemstate, considerthe following statetransitionexample. Supposeone day's state is "0221" and 20 extendedattack fires. Supposealso that 5 of thosefires are extinguishedduringthe day, and fire fightershave not returnedfrom 10 new fire arrivals by 0900h the next day. The next day's extended attack fire status is therefore 25. The next day's fire load status is, say, "0230." Given the uncertainty concerning the numberof new fire arrivalsand the outcomeof suppression efforts,the outgoingextendedattack fire statusis not deterministic,but is representedby a probabilitydistribution.The structuralmodelgivesus this distribution.We describeit first with a generaloutline of the factorsconsidered,and then in more detail.

The number of regular fire fighters hired is first specified, and the resultingtransitionmatrix is calculated.A differenttransitionmatrix is produced for each regular fire fighter hiring level considered. The model has tactical rulesfor the daily hiringof temporaryfn'efighters,the assignmentof fire fighters to initial attack standbyor extendedattack fires, and the movement of fire fightersbetweenregions.It calculatesthe consequencesof the given regularfire-fighter hiring level and the tactical rules, but does not decide amonghiring or tactical alternatives. Figure 2 outlinesthe structuralmodel. The incomingstateis represented by boxes[1] and [5], and the outgoingstateby boxes[4] and [9]. The probabilities of the outgoing fire load status come from the fire load status Markov chain. The fire load status[1] specifiesthe fire load classby region [2], which determineseach region'sprobabilitydistributionof fire arrivals, and the difficulty of controllingthem. A tacticalrule specifiesthe numberof fn'e fighters required for initial attack standbyfor the day in each region accordingto its fire load class.The provincialinitial attack standbyrequirement is the sum of the regional needs [8].

The probabilitydistributionof the numberof fire fighterscommittedto extendedattack fires [6] is the other componentof fire-fighterdemand,and dependson the extendedattackfire status[5]. The total fh'e-fighterrequirement for the day [7] is comparedto the number hired for the season[10], and any shortageis filled by hiring temporaryfire fightersof varioustypes [11]. Since temporary fire fighters are less effective than regular fire fighters, an index of work force effectivenessis calculated[11], which affects the distribution of (a) fire arrivals that become extended attack fn'es [2], and (b) old extendedattack fires that continueto burn the next day [12]. The provincial number of new extended attack fires [3] is added to the number of old extended attack fires still burning [12] to obtain the outgoing extended attack fh'e status [9].

Thus, the model addressesthe use of fire fightersfor both initial and extended attack, the temporaryhiring of extra fire fighterswhen needed,and the movementof fire fightersto regionswhere they are needed. Shortages of regularfire fightersincreasethe probabilitythat fire arrivals becomeextended attack fires, and that old extended attack fires will continueto burn into the next day. The conceptualmodel describedabove could be developedand implemented using simulation, analytical models, historical data, and/or subjective assessments; it couldbe as detailedor simplifiedas required. To clarify 652/FOREST SCIENCE

[1]

[s]

EXTENDED FIRE LOAD A•¾ACK FIRE

STATUS TODAY STATUS TODAY

[6] i

[2]

NO. OF FIRE FIGHTERS ON EXTENDED A•¾AC K FIRES

BY REGION: -

FIRE LOAD CLASS

-

NEW FIRE ARRIVALS

-

HIRING

LEVEL

[11]i

[• TEMPO TOTAL RRE [8] •__ REQUIREMENT .•-DIFFICULTY

REQUIREMENT

HIRING, WORK FORCE

TODAY

EFFECTIVENES •.

FIGHTER

OF

CONTROL

-

SEASONAL

INITIAL A1-FACK

STANDBY

NO. OF FIRE FIGHTERS ON INITIAL ATTACK STANDBY

NEW EXTENDED A1-FACK FIRES

[3]•

[12]

!!

EXTENDED I NEW EXTENDED OLD A'FfACK FIRES •

A1-FACK FRES

[4] '"'-

STLLBURNING!

[9] FIRE LOAD STATUS TOMORROW

EXTENDED A1-FACK FiRE STATUS TOMORROW

FIGURE 2. Illustration ofthestructural model thatdetermines theoutgoing extended attack fire statusprobabilities.

theconceptual model, andto showthedatauponwhichtheresults are based, weoutline therepresentative hypothetical datainthesimple structural model we used.2

REPRESENTATIVE HYPOTHETICAL DATA

A simple procedure wasused togenerate afn'eloadstatus transition matrix. Thefollowing transition matrix wasassumed torepresent thefn'eloadclass transitionsfor eachregion.3 2A moredetailed description witha numerical example isavailable fromtheauthors.

3It wasadapted fromunpublished work byD. L. Booth attheFaculty ofForestry, University of Toronto.

SEPTEMBER 1988/653

.7708 .2092.0200.0000

[

ß2777.3651 .3175.0397 ß2427.1554.4466.1553 ß1786.1071 .4643 . 2500

where the order of fire load classesis low, moderate,high, and extreme. We assumedthat each region's transitionswere independentof the others'. The number of fire fighters committed to each extended attack fire is assumedto be an independentrandomvariablewith a mean of 15 and a variance of 100. The total provincialcommitment,C, is the sumof these, and was assumedto be normallydistributed.The continuousnormaldistribution

is thendiscretized intofive values,cid = 1.....

5. For eachincoming

state, each of these five values is used in turn in the following calculations. The number of fire fighters assignedto initial attack standbyin region r,

Sr, is givenin Table1. Thetotalfirefighterrequirement for theday,•Sr + cj, is compared to thenumber of regularfirefighters hiredfor the season, and any shortageis filled by temporaryhiring. Shortagesare filled first from a pool of 150 "auxiliary fire fighters" (trained employeesfrom the government and forest industry), and then from an unlimited pool of "EFF fire fighters" (trained membersof the public). Auxiliary and EFF fire fighters were assumedto be 0.75 and 0.40 as effective as regularfire fighters, respectively. The index of work force effectivenessis W = (Reg + 0.75 Aux + 0.40 EFF)/(Reg + Aux + EFF) where Reg, Aux, and EFF are the numbersof regular, auxiliary, and EFF fire fighters, respectively. The number of fire arrivals in region r is Poissondistributedwith param-

eter •kr that dependson the fire load class(Table 1). This is a reasonable assumptionthat has some theoretical, and empirical justification (Cunninghamand Martell 1973). We assumedthat the probability that a fire arrival becomesan extendedattack fire, Pr, is the samefor all fire arrivalsin region r, and dependson the fire load class.Whether each fire arrival becomesan extendedattack fire or not is assumedto be independentof what happensto the other fire arrivals. Each fire arrival in a region can becomean extendedattack fire due to the queueingof fires to wait for initial attack resources,difficultyof control, or both. The probability of the first event is

TABLE 1.

Values of modelparametersbyfire load class. Fire load class Low

Moderate

High

Extreme

Number of fire fighters on

initial attack standby,S, Poissondistributed, expected numberof fire arrivals, h, Difficulty of control parameter a• in Equation(1) Difficulty of control parameter br in Equation(1)

654/ FOREST SCIENCE

10

45

75

135

0.4

3.4

7.4

16.0

0.97

0.88

0.77

0.64

0.50

0.75

1.00

2.00

Pqr= exp{- WSr/(5•r) •. While based on the ratio of effective five person crews to the expected numberof fire arrivals, it is an arbitrary functionthat producesthe desired probabilities.The probabilityof the secondevent is

Pdr= 1 -- arwbr,

(1)

wherethe constants ar andbrare givenin Table1. The probabilitythata fire arrival becomesan extendedattack fire due to either cause,assumingindependence, is

Pr = Pqr + Pdr- PqrPdr. The number of new extendedattack fires from region r is Poissondistributed with parameter•'rPr, and the total numberfrom all regionsis Poisson

distributed withparameter •r3'rPr' The probability that an extendedattack fire is still burningthe next day, Ps, is assumedto be the samefor all extendedattackfires, and dependson the severity of the fire load status. Whether each extended attack fire is extinguishedor not is assumedto be independentof what happensto the

otherfires.Theindexof fire loadstatusseverityis XrX r; a surrogate indicator of fuel moisture conditionsand consequentdifficulty of control. The index rangesfrom 2 to 80. It was assumedthat (for regular fire fighters) when the index is 2, ps = 0.15, and when it is 80, Ps = 0.85. The relationshipfor intermediate values is assumedto be linear. Adjustingfor work force effectiveness,

If the incomingextendedattack fire statusis ni, then the number of old extendedattackfires still burningthe next day is binomiallydistributedwith parametersni, and p•. The distributionof the numberof extendedattack fires burning tomorrow is obtained by a numerical convolution correspondingto the sum of the numberof new extendedattack fires, and the old ones still burning.

Table 2 summarizeshow the above stepsgive the conditionalprobability

TABLE 2.

Summaryof stepsin calculatingsystemstate transitionprobabilities.

DO OVER all incomingstates: Calculatediscretedistributionof numberof fire fighterscommittedon extendedattack fires, C.

DO OVER all discretevaluesof committment, c.•: Calculateinitial attack standbyrequirement,and total requirement.Hire if necessary,and calculate work force effectiveness, W.

Calculate distribution ofnewextended attack fires(Poisson - • k,pr). r

Calculatedistributionof old extendedattackfiresstillburning(binomial- n•,Ps). Calculateconditionaldistributionof numberof extendedattackfires burningtomorrow,

given% (Convolution of abovetwodistributions for thesumof therandomvariables.) Calculateunconditional distribution of outgoingnumberof extendedattackfires,No. Calculatejoint distributionof outgoingnumberof extendedattack fires and fire load status.

SEPTEMBER 1988/655

of No, the numberof extendedattack fires tomorrow,given the extended

attackfire commitment, c•. Twofurtherstepsgivethe requiredtransition probabilities.First, the unconditionaldistributionof No is calculatedby

Pr{No= no}= Zi Pr{No= noI C = cj}' Pr{C = cj}.Second, thejoint

distributionof the outgoingfu'eload statusand extend6dattack fire statusis needed.Sincethe individualstatustransitionsare independent,the probability of the joint state transitionis the product of the probabilitiesof the individualstatustransitions.Finally, the abovecalculationsare repeatedfor all incoming states to produce the complete transition matrix for the selectedregularfire fighterhiringlevel. We stressthat the simplifyingassumptions of independence,equalprobabilities, and Poissondistributionsare not essentialto the model; they were made to allow the probabilitydistributionsto be combinedanalytically. n Less restrictive assumptionswould only require that the distributionsbe combined numerically. EVALUATION

OF SEASONAL

HIRING

LEVELS

System transition matrices were producedfor regular fire fighter hiring levels rangingfrom 200 to l100, and they were assumedto apply for an entire 123-dayfu'e season.A standardcomputationon the transitionmatrices yields the probabilitythat the systemis in each state each day. Transientprobabilitiescloselyapproximatedthe steadystateprobabilitiesafter 10 to 20 days, dependingon the regularfire-fighterhiringlevel, so the cost evaluationwas basedon the steadystateprobabilities. Each state, and value of c: within the state, has a cost related to regular and temporaryfu'e-fighterhmng, and an extendedattackfire costplusloss. No costs are associatedwith state transitions.Presuppressioncosts (other than for regularfire fighters)were assumedto be constantover the relevant rangeof hiringlevels,andwere thereforeignored.Initial attackand standby costswere assumedto be independentof the regularfire-fighterhiringlevel, and were also ignored.The followingrepresentativehypotheticalcostswere used. HIRING

COSTS

A regular, auxiliary, and EFF fire fighter on initial attack standby costs

$100,$110,and $90per day,respectively, representing regulartime wages, benefits,and basicequipment.An extra $20per day is incurredfor eachfu'e fighter committedto an extendedattack fu'e. When a regular fire fighter is not needed,the daily cost is $50 which representsthe net cost after deductingthe value of his or her employmenton nonfu'ework. EXTENDED

ATrACK

FIRE COST PLUS LOSS

By this we mean the "expected daily cost plus expecteddaily present net loss." A simpleanalysisof aggregatestatisticson suppression costsand a 4 Due to concernaboutthe computational burden,we programmeda somewhatdifferent extendedattackfire status:it wasthe parameterofa Poissondistribution,whichrepresentsthe probabilityof 0, 1.... fires. Under the assumptions, whenthe incomingstatusis Poissonwith parameterO.k,the outgoingdistributionis alsoPoissonwith parameterP.kPs+ ,•X,pr,eliminatingall numericalconvolution.The outgoingdistribution parameterwascategorized intothe two closestof sevendiscretestatuses, makingthisan approximate method.A detaileddescription is availablefrom the authors.Due to the low costof computation,its use in future work is probably not necessary.

656/FOREST

SCIENCE

rough estimate of property loss and nondiscountedstumpagevalue of timber loss showedthat $6,000per day per fire was a reasonablevalue. We presentthe computationalresultsfor a broad range aroundthis value. The extended attack fire cost representsvariable extended attack fire suppressioncosts, excludingregular time fire-fighter cost and the $20 premium, which are accountedseparately.The suppressioncosts should include (1) overtime wages; (2) transportationand air tanker costs (for seasonally hired aircraft, only the costs that exceed the daily minimum charges);(3) temporary,nonfire-fighting worker costs;(4) equipmentcosts (operation,losses,retrieval, repair, and wear); and (5) extra prevention,detection, communication, and administration costs. Ideally, extended attack fire loss should include market and nonmarket;

direct and indirect; and presentand discountedfuture net costsand losses

to the peopleof the province(Althausand Mills 1982,Mishan 1982,Rezende 1982).It shouldincludethe net lossesof timbervalue, disruptedeconomic activity, property, soil erosion,water quality, air pollution, esthetics, and wildlife habitat(Marty and Barney 1981,Gorte and Baumgartner1983). We generallyagreewith the frameworkof Althausand Mills (1982)except we feel that the substitutionof goodsfrom elsewherein the provinceshould be considered(Simard1976).A comprehensive economicanalysisof multiple resourcevalueswould be neededto providea value of extendedattack fire loss for decision-makingpurposes.Mills and Flowers (1986) evaluated the effect of fire on timber valuesand reviewedpreviousresearchon the problem. We used the objectiveof minimizingtotal cost plus loss, or more precisely, the annual expectednet regular and temporaryfire fighter hiring cost, plus extendedattack fire costplus loss. We calculatedtotal costplus lossfor 200, 300, 350, 400, 450, 500, 600, 800, and 1100regularfire fighters. To test the sensitivityof the optimal solutionto the extendedattack fire cost plus loss, we used four values of the latter: $3,000, $5,000, $7,000, and $9,000per day. Figure3 showsthe total costpluslossas a functionof the number of regular fire fighters hired for the four values of extended attack fire cost plus loss. The optimalhiring level for the four valuesis approximately 450, 490, 510, and 530, respectively. The optimalhiringlevel is relatively insensitiveto the extendedattackfire cost plus loss, and hence,even less so to just the losscomponent.This is partly due to the use of fixed tacticalrulesthat specifyfire-fighterstandby requirements and extended attack fire commitments.For low extended attack fire cost plus loss values, fewer fire fighters could be economically justifiedfor theseuses.The fixed rulesare intendedto be applicablein the vicinity of the actualcostplus loss, and would give distortedresultselsewhere. Primarily, however, the solution is dominated by the tradeoff between regular and temporary hiring costs. The generouscost recovery schemefor surplusregularfire fightersincreasesthe optimum solutioninto a range where the expectednumber of extendedattack fires happensto changerelativelylittle with the regularfire fighterlevel. With no cost recovery, the solution would be more sensitiveto the extended attack fire cost plus loss. The modelwas usedto roughlyestimatethe value of centralizedprovincial controlof fire fighters.In the originalmodel,fire fighterswere freely moved at no cost to where they were needed. In a revised version, the oppositeextreme was invoked: fire fighters were not moved between reg•onsfor initial or extendedattack, the latter thus being distinguishedby region of origin. In a sense,it is a measureof the economyof scale,where SEPTEMBER 1988/657

24'

22'

20'

(• •

\ ' ',

,



•,

EXTENDEDA•ACK FIRE COSTPLUSLOSS{S/DAY} •

.• 9000

\•

. •,'

..O 7000

",, •%a ...Er' -Et_E)_.__.O--"' •',

0

200

..,•5000

.-•'....

400

600

800

3000

1000

1200

1400

NUMBER OF REGULAR FIRE FIGHTERS HIRED

Fioum• 3. Total cost plus lossas a functionof the numberof regularfire fighters hired, for four values of extended attack fire cost plus loss.

the size of the organizationdeterminesthe extent of fire-fighter sharing. Figure 4 showsthe value of the objectivefunction, and the optimumhiring level, as a functionof the extendedattack fire costplus loss,for the original and revisedmodels.With centralizedcontrol, total cost plus lossis about 30% lower, and the optimal hiring level is 30%-to 50% lower. These values are upper limits, because no province operates at either extreme. Fire fighters move fairly freely between regionsin Ontario, but not in all other Canadian provinces. DISCUSSION VALIDATION

We validated the model by subjectivelyevaluatingthe reasonablenessof its behavior and solutions.Since hypotheticaldata were used, we did not attempt to evaluate its assumptionsand predictionsempirically.Real data, 658/FOREST SCIENCE

24'

WITHOUT CENTRA

22

20'

CONTROL

z

o

o

600

o

.//

/O'"" "WITH CENTRALIZED CONTROL

/'/' 450 o

400

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EXTENDEDATTACKFIRECOSTPLUSLOSS(S/DAY) FIGURE 4. Value of objective function and optimal hiring level, versus extended attack fire cost plus loss;with and without centralizedcontrolof fire fighters.

and supportinghigherand lower level modelswouldhaveto be usedbefore the model could be more fully validated and implemented.This model requires an estimate of the value of fire loss from higher level economic models.It alsorequiresdetailsof regionaland extendedattackfire suppression activities and capabilitiesfrom lower level models.The justifiable degree of refinement of all of the modelingwould be governedby the effect on the optimal solution. MINOR EXTENSIONS

We briefly enumeratea few extensionsthat can be made without greatly alteringthe model's structure. 1. More classesof regularand temporaryfire fighterscouldbe distinguished,includingthoseborrowedfrom outsidethe province.Trainingandforeignbasing costs could be added.

2. The numberof fire fightersassignedto initial attack standbyand extendedattack

SEPTEMBER 1988/659

fires could dependon the state, as could fire extinguishmentprobabilities,and extendedattack fire cost plus loss. Regularfire fighterscould be preferentially assignedto initial attack. 3. Minimum hiringdurationsfor temporaryfire fighters,and other schedulingrules such as mandatory and regular days off, cannot be accurately modeled due to the memorylesspropertyof the Markov chain. An approximationfor days off is possibleby assumingthat they are uniformly scheduledover the season. 4. Fire-fighter requirementsfor a prescribedburn program could be included. 5. Extended attack fires could be distinguishedby region of origin and size category by expandingthe state space. TRANSPORTATION COST

The transportationcost relevant to this decisionis the cost of movingregular fire fightersbetweenregionsin responseto changingregionalfu'eloads, which increasesas the numberof regularfire fightershired decreases.5The aggregatecost of all repositioningin Ontario is small comparedto the cost of the regularfire fighters, so ignoringthe transportationcost is not crucial. Much of that cost, however, could be captured by assigning a cost to changesin the fu'e load status that cause fu'e fighters to be moved. This would only miss costswhen a fire load statuslike "0401" stayedthe same, but the actual extreme region changed.Alternatively, all the regions could be made distinct, or the relocationdecisionscouldbe cost evaluatedexplicitly as in Greulich's (1976) Markov decisionprocessmodel. OTHER SUPPRESSION RESOURCES

Our model addressesonly part of the decision-maker'sproblem, which includes choosing the presuppressionlevel of all fire-fighting resources. Based on cost and effectiveness,we believe that transporthelicoptersand air tankers are our most seriousomissions.In any case, the initial attack and extended attack probability submodelscould in principle consider mixes of fire suppressionresources.This mightbe done usingthe approach of FEES, but with queueing caused by multiple simultaneousfire occurrence incorporatedas in Martell et al. (1984). Then a searchwould be conductedfor the optimal level and mix of all resources. LITERATURE

CITED

ALTHAUS,I. A., and T. J. MILLS. 1982.Resourcevaluesin analyzingfire managementprogramsfor economicefficiency.USDA For. Serv. Gem Tech.Rep. PSW-57.9 p. ANEJA,Y. P., and M. PARLAR.1984.Optimalstaffingof a forestfire fightingorganization.Can. J. For. Res. 14:589-594.

BRATtEN, E W., et al. 1981. FOCUS: a fire managementplanningsystem--final report. USDA For. Serv. Gem Tech. Rep. PSW-49. 34 p. CUNNINGHAM,A. A., and D. L. MARTELL.1973.A stochasticmodelfor the occurrenceof man-causedforest fires. Can. J. For. Res. 3(2):282-287.

DOAN, G. E. 1974.Optimal initial attack systemdesignfor forestfire control. M.Sc.F. thesis, Fac. For., Univ. Toronto. 140 p. + app.

5 The cost of transportingall fire fightersand equipmentto, from, and around extended attack fires is included in the extended attack fire cost.

660/FOREST

SCIENCE

GAMACHE,A. E. 1969.Developmentof a methodfor determiningthe optimumlevel of forest

fire suppression manpower on a seasonal basis.Ph.D. diss.,Univ. Wash.,Seattle.164p. GORTE,R. W., and D.C. BAUMOARTNER. 1983.A fire effectsappraisalsystemfor Wisconsin. USDA For. Serv. Gen. Tech. Rep. NC-90. 8 p. GREULICH,F. E. 1976.A modelfor the seasonalassignmentof air tankersto homebasesunder optimalexpecteddaily transfer/userules.Ph.D. diss.,Univ. Calif., Berkeley.92 p. + app. HOWARD,R. A. 1971.Dynamic probabilisticsystems.Volume I: Markov models.JohnWiley and Sons, New York. 577 p. + ind. MARTELL, D. L. 1982. A review of operationalresearchstudiesin forest fire management. Can. J. For. Res. 12(2):119-140. MARTELL, D. L., et al. 1984. An evaluation of forest fire initial attack resources. Interfaces 14(5):20-32.

MARTY, R. J., and R. J. BAR•E¾. 1981.Fire costs,losses,and benefits:an economicvaluation procedure. USDA For. Serv. Gen. Tech. Rep. INT-108. 11 p.

MILLS, T. J., and E W. BRATTEN.1982.FEES: designof a fire economicsevaluationsystem. USDA For. Serv. Gen. Tech. Rep. PSW-65.26 p. MILLS, T. J., and P. J. FLOWERS.1986. Wildfire impactson the presentnet value of timber stands:illustrationsin the northernRocky Mountains.For. Sci. 32(3):707-724. MISHAP, E. J. 1982.Cost-benefitanalysis.Ed. 3. GeorgeAllen and Unwin, London. 447 p. ONTARIOMINISTRYOF NATURALRESOURCES. 1986.Statistics1986--a statisticalsupplement to the annual report of the Minister of Natural Resourcesfor the year endingMarch 31, 1986.Queen's Printer for Ontario, Ontario, Canada. 153p. QUrNTILIO,D., and A. W. ANDERSON.1976.Simulationstudyof initial attack fire operations in the WhitecourtForest,Alberta. Environ. Canada,Can. For. Serv. Inf. Rep. NOR-X-166, 35 p.

REZENDE,J. L. 1982. Applicationof benefit-costanalysisto forestryinvestmentproblems. Ph.D. thesis, Fac. For., Univ. Toronto. 190 p. SI•ARD, A. J. 1973.Analysisof a simulatedsupplementalairtankertransfersystem.Environ. Canada, Can. For. Serv. Inf. Rep. FF-X-41.40 p. SIMARD,A. J. 1976.Wildlandfire managementthe economicsof policy alternatives.Environ. Canada,Can. For. Serv. Forest. Tech. Rep. 15.52 p. WAONER,H. M. 1975. Principlesof operationsresearch. Ed. 2. Prentice-Hall, Englewood Cliffs, NJ. 1039p.

SEPTEMBER 1988/661

A Markov Chain Model for Evaluating Seasonal Forest Fire Fighter ...

ABSTRACT. A model was developed to help resolve the decision of how many fire fighters a large forest fire management agency should hire for a fire season to minimize expected cost plus fire loss. It addresses the use of fire fighters for both initial and extended attack, the temporary hiring of extra fire fighters to satisfy ...

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