64 S. Ríos-Insua et al. / A time-dependent decision support systemfor multi-attribate decision-making
equal to zero to any fixed or given attribute that is notrelevant in a particular time period, and its contribu-tion is then null. This is implemented in the systemby means of attribute activation/deactivation. Figure 1shows a diagram of such a hierarchy. Intertemporaldecisions could be considered as an explicit trade-offbetween the present and the future by structuring theobjective tree so that the importance of the significantattributes in each period is made explicit.
The DSS is based on an additive multiattribute utilitymodel [16], which allows for imprecision concerningthe inputs Í2Il and[24], andpartial information of theDM's preferences. Thus, the process of assessing theindividual utility functions and the constant scales isnot very demanding, which is less stressful on DMs.
Furthermore, we shall consider the situation wherethe consequences might have some uncertain policy ef-fects, as considered in [18]. So, the system will alsoallow for uncertainty about the consequences or out-comes, where they can be entered as ranges or intervalsinstead of single values as an approach under certaintywould demand. Note that this more general viewpointleads to a more robust approach to decision making,which can account for imprecision and could overcomesome of the criticisms of Decision Analysis.
Thus, the starting point will be to establish a set of nattributes denoted by Xt, ..., X, and the time periodst : 1,...,,7. Then, the consequence of each strategyor alternative decision ̂ 9s e S, where S is the availablestrategies set, is a stream defined by a vector of intervals
where rlj nd r{nt are, respectively, the lower (,0)and upper (t/) endpoints of the imprecise consequencefor attribute X¿ in the time period ú. We assume thatthere is a continuous uniform distribution over eachinterval l"li,rYn'f, and both endpoints being equal,i.e., r!] : rYst, would be equivalent to the case undercertainty, where the policy effects for a strategy ,Sq inattribute X¿ for time periodt ne precisely known. Inany case, we also considerthepossibility of substitutingeach interval by a single value given by the average,f,nt : @l] + "Yn') 12 (P means precision), havingthen a precise consequence, if deemed appropriate bythe DM.
Next, an analysis focused on judgemental inputsmust be conducted to assess imprecise utility functionson atfributes and imprecise scaling factors or weightson objectives and atfributes in the objective ffee, and
for their aggregation into a global utility function thatincludes, where appropriate, discount factors for eachattribute set to evaluate timestreams. as we shall ex-plain below. The methodology has been implementedon a PC-based DSS, where all process-relevant infor-mation can be entered to help DMs arrive at the best ora satisfactory strategy. Finally, a multiparametric sen-sitivity analysis is introduced to check the sensitivityof conclusions on the inputs.
The following deals with the multiattribute utilitymodelling process under imprecision. Section 3 con-siders different discounting methods for providing DMswith assistance concerning how to tradeoffpresent andfuture. Due to the diffrculty in determining the beststrategy for problems with a large number of them,an approximation approach is proposed in section 4,supported by a optimizationmethodology based on in-teractive multiobjective simulated annealing, which isdescribed in section 5. Section 6 outlines multipara-metric sensitivity analysis. In section 7, we show thefinal model formulation illustrated with an example onrestoring aquatic ecosystems and, finall¡ some conclu-sions are provided.
2. L procedure for assessing the imprecisemultiattribute utility function
Expected multiattribute utility theory can be consid-ered as a leading paradigm for normative decision the-ory. However, multiattribute utility theory calls for theDM to provide all the information describing the deci-sion situation to assess component scalar utility func-tions u¿ 16,251, and [28]. This can be far too strictin most practical situations, which could lead to theconsideration of imprecise component utility functions,see [23] and [30].
In this modelling context, what would be called im-precise timestream utility fficient strategy set playsan important role, because it has a similar property tothe well-known efficient or Pareto optimal strategy set:given a strategy in this set there is no other strategy in Sthat dominates it. However, this set can be diffrcult todetermine in this new context, so intelligent approachesare needed to help the DM to arrive at a final solution,as we shall see later.
Now, note that it is necessary to assess a scalar util-ity function u¿ for each attribute X¿, that reflects DMpreferences on the possible atffibute values. The draw-backs associated with utility function assignment arewell known, even though good software is available for
s9?un¡aa-uolnlrap anqluto-ulnu nlualsKs uoddns uotsnap Tuapuadap-awg V / 'to P Dnsul-soJü'S
66 S. Ríos-lnsua et al. / A time-dependent decision support systemfor multi-attribute decision-mnking
Class of utility ñrnctions withthe extreme gambles method The intersection
of both classes
Class of utility functions withthe fractile method
would have provided inconsistent responses and he/sheshould reassess his/her preferences. Thus, the intersec-tion will be the range for the DM's utility functions. Thesystem is able to detect possible inconsistencies andsuggests what the DM should change to achieve con-sistency. Thus, given a precise consequenee fi¿ for at-tribute X¿,wehave a utility interval l"! ("u), uY @o)linstead of a single utility u¿ (r).This is shown for in-creasing utility functions in Figs 2 and 3 by the stripedarea.
As we shall see below, the evaluationprocess calls forprecise utility functions in the problem-solving process,so the system assesses fitted utility functions "f emeans precision) by natural cubic splines interpolationthrough the mid-points of the utility intervals in theintersection area for each 'u,¿, see Fig. 3. The result fordecreasing utility functions would be similar.
We also need to assess positive scaling constants orweights k¿ to add the separate contributions of the at-tributes and get the additive multiattribute utility func-tion. The DSS includes two methods for assessingsuch weights for any objective of the hierarchy: usingtradeoffs, see [16], and direct assignment. The firstprocedure is based on trade-offs among the respectiveattributes at the lowest-level objectives stemming fromone and the same objective. The DM is asked to pro-vide probability intervals such that he/she is indifferentbetween lotteries and sure consequences. This proce-dure is best suited for the lowest-level objectives in the
Fig.2. Utility function classes of based on the fractile and extreme gambles methods for attribute -{.
hierarchy because they involve a more specific area ofknowledge. We begin with the objectives at the low-est level ofthe hierarchy and then continue the assess-ment in ascending order through the hierarchy. As inthe case of utility assessment, imprecise assignmentsby means of ranges or intervah fkr¿, k{] arepossible.This means that the DM will be able to perform a globalsensitivity analysis, allowing intervention at any levelof the objective hierarchy.
The second procedure is based on direct assessmentand is, perhaps, more suitable for upper level objec-tives, because they can be more political. Here, theDM is asked to provide weight intervals lk!,k{1, asbefore. Note that when the system is run, the startingpoint is equally weighted objectives. However, any in-terval or precise weight can be changed, and the systemautomatically takes care of how these changes shouldbe propagated through the objective hierarchy and re-calculates the overall vector utility for each strategy.
In both assessment procedures, the system com-putes the normalizedaverage weights and a normalizedweight interval over the directly assigned weights orweights outputby taking expectedutilities in the lotter-ies and sure values in the assignments based on trade-offs. In any case, such normalized average weightstaken from the DM's imprecise responses will be de-noted bV kf . The superscript P in kl means preciseweights, and note that the DM can specify identicalinterval endpoints, which means thatthe DM is consid-ering a precise value, i.e., kf; - ky - kf .
'ta enqu¡e JoJ uorlcunJ ,{Upn eslce¡d pue sse¡c uorpurg ,{lruln '€ '8Id
H
(ix¡ )n
,X
Tn
?ut>trut-uotsoap aflqugo-u1nut tolutatsds uút rcl4np ruzpuadap-au4t V / 'to ta onsq-so!ü 'S
l
68 S. Ríos-Insua et al. / A time-dependent decision support systenxfor multi-attribute decision-mnking
rameters r¿ &te the social discountrates and the expres-sion 1 *r¿ - pl-' I ptrepresents the trade-offs betweenthe costs and benefits in any two adjacent time periodsú - 1 and ú. The problem of determining the appropriatesocial discount rate has received much attention, see,e.s . l2 l .
For medium to long time spans, i.e. for large T, pr-tbecomes very small as ú tends to T, which means thatconsequences towards the end of the timestream areeffectively discounted to zero. To overcome this objec-tion, we have also implemented an approach proposedin [10] and [11], which suggest a varying discountrate that could devaluate the future at a much slowerrate. This is the p ro p o rti o nal - dí s c o untin g mo d el,wherethe timing weights form a linear fractional sequence,that is, (bol (bo + t))", where b¿ ) 0 is the tempo-ral mid-value for attribute X¿, that is, the point inthe future given half the weight of the present, and-oo ( r¿ 1 x reflects how the ratio of the outcomesis related to the ratio of the time effects. So, if theDM is timing averse, the utility of a future stream ofimprecise multiattributed outcomes is
l T r L T , . .
lD, ni "i {"iil+\L,f #),, k! u! (" o"h,L ¿ : t i : 2 t : L
v x I v
G )T n T h . I
D, nY "Y ("fln, )+f t( r= ),, kY "Y @Yh It : l , i :2 t :L
vL I v I
Note that the values in expression (4) decay to 0 astime tends to infinity slower than (3) and, therefore, thisapproach attaches relatively more importance to long-term outcomes. The model focuses on the ratio betweentwo time periods and not on the absolute difference intheir time. To use model (4), it is necessary to determineb¡ and r¿ for each attribute X¿ for the DM, see [10]and [11].
The next thing to be modelled in the DSS is the casewhere the DM assumes that the assessment of discountfactors is a diffrcult task for certain scenarios, and analternative approach must be provided. In this situation,we consider for each strategy ,Sq e S, the vector ofutility intervals (2) for each time period ú and, as before,the problem involves choosing the best strategy nowbased on that vector.
In short, we have that each strategy ,Sq will be char-acterized in all the above cases by a vector of utilityintervals
o (Sn) : uq: ( l " | t , " f l t l , . . . ,1uf ; ' , "Y\)of dimension 1 or T, depending on the case. Next,we provide a problem-solving approach to solve thiscomplex vector optimization problem.
4. A problem-solving approach based onapproximation
First we note that a preference relation Fo can bedefined on S from the utility vector (2), which leads toa dominance principle, defined as: given two strategiesSs, S* € S, we have that
Sq Fo S* e un ) ar,-,
which means that Sq dominates,Sm if and only If ult 2uff,Vt, with at least one strict inequality. The relationF,, is a strict partial order on S and, hence, we state thevector optimization problem as
max ,, (,9n)s.t. ,9q e S
A natural concept is that ,9q e S is an imprecise utility
fficient vector strategy if there is no ,9- e S such thatS^ l, ,Sq or, equivalently, a* 2 u.q. This strategyset will be called imprecise timestream utilíty fficientvector set and denoted by e t (S, o). This leads to theproblem "Given S and to, find e¡ (S,u)". Clearly,if the set E¡ (S,o) had a single element ga, it wouldbe the most preferred strategy for the decision-makingproblem. However, this is not the case in most realproblems, because e¡ (S, u) could crments. Thus, our problem should O"?:#a single element from the set e¡ (S, u)". One way tosolve this decision-making problem, favored by behav-ioral approaches, will be possible if the DM is able toreveal more information on hislher preferences to pro-vide additional structural assumptions and get a subsetof e¡ (S, u). For this purpose, we provide an interactivemethod, based on multiobjective simulated annealing,to progressively build an approximation set to e7 (S, u)in collaboration with the DM, which is valid for any ofthe above settings and based on vectorial optimizationfor the imprecise utility vectors with different dimen-sions depending on the model used.
5. Interactive simulated annealing-based search
The ideaof themethod we proposeis based onthe in-teraction with set S, which is associated with the respec-tive set of vectors of utility intervals (l-dimensionalfor cases (3) and (4) and ?-dimensional for (2)). Wethen generate an approximation set A (S, o) of the im-precise timestream utility efficient vector set E¡ (S, o),i.e., solutions not dominated by any other consideredsolution.
-na se8ueqc osoq1 ̂doqlo seJ€c se>I€lure1s,(s eq1'1elre1ur lqErert pezqeuuou € Jo punoq e ro lq8rem pezr
lndlno oluosueduroc esll\rr€dr(q (V 'n'S) y,&\fY 0leseql rellg ol peeu em "{1¡uurd 'u ,{q pscnpq uo4ce4p eql
uI suortnlos ,(l¡pn luercule,(1¡epuelod eq] sulsluoJ lerp(y'n'S) V lsll u 'y IIme roJ 1e8 e¡t ',{elt peglsre^Ip,{¡elrsuelxe uu uI pelureueE 'ÁA les Jolce 8upq8¡eme 8urr1e¡ :s^\oIIoJ se sr'(n'S) rf les eqtJo (n'S )Vuouaruxordde pooE e Sururelqo JoJ ernpecord eq¡
'suollnlos
,(1¡¡pn luercgo,(¡er1ue1od eqt,{q peuroJ uor8e¡ JBIncrl-¡ed e spre^\ol sn p€el plnoc rcqt slq8le^{ Jo les ursuece 8ur1celes sueelu qclq^\ 'suorln¡os ̂\eu Jo ,(1¡¡quqord
lueurrre^ourrr eql ol Eurprocce ',(ern srql uI '{41gqoqoñ
acuoldaccp w u1(ou{'r$¡gquqordue,rrE 3 qll^\ uopeJell
lxou eql JoJ ,g Suueprsuoc enu4uoc plno,¿l\ em '(n'S) V
uI lueluele ,(ue ,(q pepu¡tuop ere^\ s JI 'puBt{ reqlo
oqt uO 'S Áq peteulruop (.'S) V ur uo4nlos ,(ue 1no^\orql pue (n'S)V tes eqt ot S ppu e^\'essc slt{l uI'les uorlerurxo¡dde eql q ,(llueunc set8ele.us eql ,{qpeleurruop ]ou sI U JI peldecce sr ,S pue 'pereprsuoc
sr uorturelr luerrnc eq] q uoqnlos eg] Jo pooqroqq8teueql Iuoq g ,{3ep4s Jeqloue 'suoqe¡ell Surmo¡o¡ eqtuI 'oS ,(po Surureluoc pezrl€rlrur q (n'S)V tos eqlsnql pu€ 'ruopuel 1B u^\?rp s ) os ,(8ele¡ls leplul ueEurpr,rord uoIlBJeU lsrg e qlrm sur8eq poqloru eql
'e¡durs sr poqlelu eql Jo eepl crseqeql 'uer{,L '(.'S) /f tes eqt Jo (g ur ser8epüs eq} qtl^\pep}?rcosss srolre^ ,$IIpn eql stueserder n) (o's) y ps
uouúu)rxotddo ue elereue8 ueql o [ 'Dn Jo]col ,{tnlne qlF\ pelsrcosse sl ¿,9 lueruele qffie ereq^\ 's les eqluo peseq sr poqletu eqJ 's^\oIIoJ se s '[91] eQs'uor1-anauotddo uo pasDq Sutloauuo papputs autJuJalulpolpc 'pogtetu uollen¡e e slggo eepr Eur,{¡repun eql
Swow-uuwap atnqyuD-ulnu n{waLsKs ltoddns uotsnaP Tuapuadap-aug V /'1CI P Dnsq-so!ü 'S
70 S. Ríos-Insua et al. / A time-dependent decision support systemfor multi-attribute decision-mnking
gated through the objective hierarchy and automaticallyrecalculates the list -Ls of potentially optimal solutionsto run a new interactive problem-solving process. Italso recalculates the overall utility for each strategy andthe resulting ranking based on the precise values for theweights, individual utilities and consequences.
Another way of performing SA involves assessingthe interval in which the weight for a specific objec-tive can vary, maintaining a constant ratio among theother weights, without affecting the overall strategyranking. Suppose that we currently have a ranking ofthe given strategies and the DM chooses a node or leafof the objective tree with an associated weight. Thesystem calculates the weight interval for this node/leaf,taking into account the updated weights for the objec-tives stemming from its predecessor, so that the rankingbased on the precise assignments does not change. Inother words, if the weight is changed and the new valueis within the range, then the ranking will not change.However, if the new value is not within the range, thenew ranking will be different to the previous one.
Finally, the system runs simulation techniques forSA, see [3] and [4]. This kind of sensitivity analysisallows for simultaneous changes in weights and gen-erates results that can be easily statistically analyzedto provide more insights into the multiattribute modelrecommendations. We propose selecting weights atrandom using a computer simulation program so thatthe results of many combinations of weights, includinga complete ranking, can be explored efficiently.
Three general classes of simulation are implemented:random weights, rank order weights and response dis-tribution weights. They are described briefly below.
a) Random weíghts.' As an extreme case, attributeweights are generated completely at random.This approach implies no knowledge whatsoeverof the relative importance of attributes. In manymulticriteria settings, the scores of the strategiessignificantly limit the subset of potential rankings.
b) Rank order weights: Randomly generatingweights while preserving an attribute rank orderplaces substantial restrictions on the domain ofpossible weights that are consistent with the DM'sjudgement of attribute importance. Therefore,the results from the rank order simulation mayprovide more meaningful results. The DM canintroduce the rank order for all or only some ofthe attributes of the problem.
c) Response distribution weighls.' The third type ofsimulation-based sensitivity analysis recognizesthat the weight assessment procedure is subject
to variation. For a single DM, this variation maybe in the form of response effor associated withweight assessment. Thus, whereas in the firstclass of simulation, attribute weights were ran-domly assigned values between 0 and 1 (takinginto account that the sum of the whole is 1). at-tribute weights are now randomly assigned val-ues taking into account the attribute normalizedweight intervals provided by the DM through theweights assignment methods.
Once the simulation has been run, the system com-putes several statistics about the rankings of each strat-egy, like mode, minimum, maximum, mean, standarddeviation and the 25th,50th and 7íthpercentiles. Thisinformation can be useful for discarding some possiblestrategies, aided by a display which presents a multipleboxplot for the strategies.
7. An application to the restoration of an aquaticecosystem
In this section, we describe the application of theDSS to the analysis of a set of remedial strategies forrestoring Lake Kozhanovskoe, located in the Bryanskregion in Russia, which was heavily contaminatedwith1379r after the Chernobyl accident in 1986. This com-plex decision-making problem was studied in a sim-pler context (taking a static approach, precise inputsand certainty over the consequences) using the MOIRAsystem, see [9]. In 1998, Lake Kozhanovskoe wasclassed as a radio-ecological reservation, and fisherywas offrcially forbidden because of the high levels offish contamination with 137Cs. The population aroundthe lake was evacuated, as its area belonged to the pop-ulation evacuation zone as the levels of contaminationwith 137Cs were rather high (137Cs fallout on the lakewas about 600000 Bq/mz). However, many residentscontinued to live at villages near the lake, and fishcaught in lake Kozhanovskoe was a predominant foodof the local residents even 10 years after the Chernobylaccident.
In spite of the ban on fishing proposed by the expertassessment,20-30 families were still engaged in fish-ing. In addition, amateur fishermen often went fishingto the lake from neighbouring districts and caught fishfor the most part with fishing rods. Consumption offish by the fishermen and members of their familiescould be as great as 80 kg a year per person, while.on average, the residents of this area consumed 20 kg
-cefqo Ielel-tse^\ol eql ol pe]ercosse selnqlr1]s eI{:[
'pesodord sporred erun eeJql eql Jo euo qcee roJ leselnqrrD€ elues eql qll^\ 'urelqord sgl roJ urer8erp eerler{l s¡aoqs y em8rg 'SSC eql q prqsllqutse erol(',(qcru-rerq eqt Jo selrlcefqo ¡elel-tsetrrol eqt qrp\ petelcosseselnqr4l€ etll se ilo,,t\ sB 'senrlcefqo lIrc ele¿ 'uo4senb
ur orJuuecs eql JoJ sel4euJelle uolluJolseJ Ieclleqlod,(qor{l Jo uor1enle^e puu uorlducsep oql roJ spunor8 eqlepr,rord ol popuelq qcH,!\ ' ure¡qord 8ur{eur-uolsrcopslrll roJ lllnq se^a ,{qcrererq errrlcefqo uy 'e I13uJe1-l€ uorsrcep uo4erolser lsoq eql Jo ocloqc sóI ic eqlsuoddns ure1s,(s uorsrcop o oqe erp lxeluoc sql uI
'uor1e¡ndod
eql ,(q porunsuoc qsu Jo Junorus eql ro uollsurlueluocqsg Jerpre Surcnper ]€ pelueuo ,(¡ureur eren serEep4suouuelJelul eq]'eJoJeJerll'serceds,fto¡eperd-uou ul
lqErerrr qse4 '8rybflrl0I-9 se,t\ ]I seereq^\ lq8real qse+'Erybflrl gZ-OT,'eBe¡e,r¿ uo 'sB^\ pue q8tq peuleruereo{snouer{zo) e{e-1 uro{l qsg Jo serceds ,fto1uperd ursJ¿er Jo uorl€Jluecuoc eql '866I uI ',(l1enuue qsu Jo
TL
'ao)ls^orreqzo) e{B'IJo uorlsJoNer eq} JoJ ,(qcmre¡q e,rqcefqo ¡uepuadep-eurll 't 'ald
Suurw-uotsnap anqlutD-ulnu rctua¡sKs ttoddns uotsnap Tuapuadap-au! V / 'Io P Dnsul-soly S
S. Ríos-Insua et al. / A timz-dependent decision sapport systemfor multi-attribute dccision-making
Table 1Attributes used in evaluating the strategies
Attribute RangeWorst z¿* tlest r;
X1 : Ecosystem IndexX2 : Dose to FishX3 : Dose to Critical IndividualX¿ : Collective DoseX5 : Amount AffectedX6 : Duration of Ban IIX7 : Cost to EconomyXs : Cost of Application
LEI @cosystem Index)mGy (MiliGray)mSv (miliSievert)
mSvxpersonTonnesMonthsEurosx 102Eurosx ld
51d500
L 2 x 1 ÚrÉ120108
2 x 1 0 8
10000000
Table2Outcome stream for S€
Attribute
XtXzXsX+XgX6X7Xg
[1.31, 1.40][1700, 1820][10.9, 11.3]
[61000,63000][540,600][60,60]
[342000,367000][o, o]
the exposure pathways due to the contaminatedwater body. It is a measure of the increased riskof serious latent health effects.
5. Amount Affected (Xs). In the case ofresfiictionsto fl sh consumption and food industry processing,the amount of fish affected by restrictions.
6. Duration of Ban II (X6). In the case of restric-tions to fish consumption and food industry pro-cessing, the duration of the restrictions.
7. Cost to Economy (Xz). The direct economicimpact of the resfrictions, either in terms of thecost of the food affected by bans or in terms ofthe production lost (e.g., share of Gross DomesticProduct Lost).
8. Cost of Application (Xa). In the case of re-medial countermeasures (physical or chemical),this represents the direct cost of the application:manpower, consumables, equipment needed forthe application, management of wastes generated,etc.
Thble 1 shows these atffibutes as well as their mea-surement units and the respective ranges provided forthe strategies analyzed later.
Several intervention strategies or alternatives mustbecreated to reflect what could happen once the scenariohas been settled. Expected conditions occur, based onthe available infonnation, but as the multiattribute con-sequences of each possible sfrategy are assumed to be
uncertain and Simulated Annealing generates stochas-tically alternatives, the system can randomly generatealternatives and any considered feasible will be takeninto account for evaluation. As a consequence, we willpossibly have a NP-hard problem that can be solvedby mathemafical algorithms. Generally, the only wayto optimize will be by simulation wittr the aid of theimplemented interactive multiobjective simulated an-nealing. As shown with this approach, DM can changeseveral parameters to interactively reduce the impreciseutility efficient set until a final alternative is obtained.
In this example, we consider six sfrategies whosedescriptions are:
SL : No Action. Natural evolution of the situationwith-out intervention.
52 : PotashTreatunent. Reduction of aquatic organismuptake by potash treatment of aquatic ecosystemscontaminated by radiocesium.
53 : Fertilization. Tonnes of fertilizer addedto the laketo increase biomass (biological dilution).
Sa : Lake Liming. Reduction of radionuclide remobil-isation from sediments.
S5 : Sediment Removat. 6k¡r.t2 of sediments removedfrom the lake down to depth of 5 cm.
56 : Autom.atic Food Bans. Automatic fish consump-tion ban when 1379. content in fish is greater than1000 Bq/kg.
Time period ú
[1 .30,1.42][1660,1800]lro.7,rL.21
[60700,62800][150,165][30,30]
[12000,48300][o, o]
[1 .28,1.43][1590, 1750][10.3, 11.0]
[60400,61300][0,0][0,0]
[11000,54500][0,0]
sa,rpcofqo lue^alo¿ 'ruC eqt ,(q pereprsuoc eq o] sorr?u-aJ6luareglp eql roJ selqeurelle uorsrcep Jo ser8ep¡lsluauaEeueur [Bcr]oq]od,(q Jo uo4enlele pue uo4ducs-ep eW roJ spunor8 eqt epr,lord ot pepuelu sr qcrq^\'a¡nlcn-qs (ae¡ e¡rqcefqo) cruruu,{p u ,{q pe¡oddns pue'sar8ale.gs elqBIIB^e eql Jo secuenbesuoc pue elnqu1leqcse roJ serlllrln pnpplpu¡ 'eer1 enlul eql Jo se¡rrlval-qo eqt IIe roJ slqSre,rr Buru¡ecuoc slndur esrce¡drur uopeseq SuueeuÉue surels.,{s ur sseco¡d srsr(¡eue uorsrceper{l roJ uels,(s uoddns uorsrcep e pedole,rep e^€q ed\
suolsnlruoJ '8
'speJJe neql lunoc3? olul e>l31 ol pesn sI ursruop
lerodurel eql JI elqulcu.B erolu eq illrr\ srue¡qord uorl-ezrunldo oseql leql sel€üsuoruep qc¡Irt\ 'EunIue¡ r(Ee-te4s oql olur lq8rsur Ieuortlppe ure8 o1 ,fi1¡ce¡ srs,(1eue,$r¡r4rsues culerueredlllnu¡ e sepnlcu S SC eql'snq¡
'ser8ele.Ds eqt roJ 1o¡dxoq pednor8,(¡zcr¡el e sluese¡d pq1,ft1dsp e Áq peplu 'ser8e1er1s
elqrssod eruos Surprecsrp JoJ Iryesn eq uec uollelrttoJulsr¡1 'sep]uec¡ed
Wgl pue q]gg 'WgZ eq] puu uorJer^-ep pJepuels 'uueru 'ruruulxeru 'urnurlulur 'epour e>lII'r{8e}e¡ls qcee Jo s8unlue¡ eql lnoqe scllsllels IBJeAesselnduroc ruelsr(s eq1 'suorlelnurs sunJ tr IC eql JI
as well as the attributes associated with the lowest-levelobjectives of the hierarchy are created in the DSS. Sce-narios need to contain enough technical informationfor them to be realistic, while they should not be sotechnical as not to be understandable. The alternativesmust be analyzed with respect to the attributes associ-ated with the lowest-level objectives of the hierarchy,some of which (maybe all) are time dependent. Next,the DM will have to assess the imprecise utility func-tions, scaling constants and discount factors, if deemedappropriate, which will be used to provide the inputsfor the coffesponding model, associating each strategywith a vector of utility intervals, whose dimension willbe equal to the number of time periods considered, sowe will have only one utility interval for the static case.The application of interactive multiobjective simulatedannealing leads to the determination of the best or, atleast, a small enough set of satisfactory strategies, fromwhich the DM could choose the final strategy more
Fig. 5. The utility intervals and the ranked strategies based on their precise utilities for each time period.
easily by changing satisfaction levels or using a mul-tiparamefric sensitivity analysis. This means that theuser can gain additional insight into the ranking of thealternatives. This demonstrates that this kind of opti-mization problems may be more tractable by using thetemporal domain to take into account their effects.
9. List of symbols
S: Set of available strategiesP: Precisionp¿ : Timing weightr¿: Social discount rateb¿: Temporal mid-value for attribute X¿Fo: Preference relation based on the vectorial utility
